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CEGEP Linear Algebra — Products of Vectors Worksheet

Q: Is the scalar product the same thing as the dot product?

Yes. The devis says produit scalaire , because the output is a scalar; English usage is usually dot product, and sometimes inner product. The cross product is the produit vectoriel , for the same reason — its output is a vector — and the triple product is the produit mixte .

By Marius — private math tutor in Montreal. Updated 30 August 2026

The geometry half of Linear Algebra opens here, and this is the set that makes the rest of it possible. What the dot product measures and why its sign alone settles acute against obtuse, how a zero dot product becomes the working definition of perpendicular, how to split a vector into a piece along another vector and a piece at right angles to it, how the cross product is built from a 3 × 3 determinant and why its output is a vector rather than a number, why swapping the two inputs flips it, what the scalar triple product says about three directions at once, and how areas of triangles and volumes of parallelepipeds fall straight out of the two of them. The last question on the sheet asks you to choose the right product without computing anything, which is the skill the rest of the course keeps asking for. Have a look on this page, then print the free PDF when you want to write on it.

Practice worksheet — free PDF

5 pages 7 questions Letter size, print-ready

Download the worksheet Read the questions here

No email, no account, no watermark. Teachers: photocopy it for your classes freely. Worked solutions and 5 harder problems come with the CEGEP Linear Algebra bundle.

All 7 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

Q1The Dot Product and the Angle Between Vectors

A drone's three guy wires leave the same anchor point along the directions $\vec{u} = (3, 4, 0), \vec{v} = (0, - 4, 3), \vec{w} = (4, - 3, 0),$ measured in metres.
1. Compute $\vec{u} \cdot \vec{v}$ and $\vec{u} \cdot \vec{w}$ .
2. Find the angle between $\vec{u}$ and $\vec{v}$ , to the nearest tenth of a degree.
3. State what the value of $\vec{u} \cdot \vec{w}$ tells you about $\vec{u}$ and $\vec{w}$ , and say how you could have predicted the sign of $\vec{u} \cdot \vec{v}$ from your answer to (b).
Q2Orthogonal Projection

Let $\vec{u} = (1, 5, - 1)$ and $\vec{v} = (2, 1, - 2)$ .
1. Find the scalar component of $\vec{u}$ along $\vec{v}$ , that is $\frac{\vec{u} \cdot \vec{v}}{‖ \vec{v} ‖}$ .
2. Find the orthogonal projection ${proj}_{\vec{v}} \vec{u}$ .
3. Write $\vec{u}$ as the sum of a vector parallel to $\vec{v}$ and a vector orthogonal to $\vec{v}$ , and verify the orthogonality.
Q3The Cross Product

Let $\vec{u} = (2, - 1, 3)$ and $\vec{v} = (1, 4, - 2)$ .
1. Compute $\vec{u} \times \vec{v}$ using the $3 \times 3$ determinant.
2. Verify that your answer is orthogonal to both $\vec{u}$ and $\vec{v}$ .
3. Write down $\vec{v} \times \vec{u}$ without computing a second determinant, and name the property you used.
Q4The Scalar Triple Product

Let $\vec{u} = (1, 2, - 1)$ , $\vec{v} = (3, 0, 2)$ and $\vec{w} = (2, - 1, 1)$ .
1. Evaluate $\vec{u} \cdot (\vec{v} \times \vec{w})$ as a single $3 \times 3$ determinant.
2. Confirm the value by computing $\vec{v} \times \vec{w}$ first and then taking its dot product with $\vec{u}$ .
3. State what your value tells you about the three vectors.
Q5Areas and Volumes from Vector Products

A sheet-metal shop marks the points $A (1, 0, 2)$ , $B (3, 2, 1)$ , $C (2, - 1, 4)$ and $D (2, 1, 5)$ , coordinates in metres.
1. Find the area of the parallelogram determined by $\vec{A B}$ and $\vec{A C}$ .
2. Find the area of the triangle $A B C$ .
3. Find the volume of the parallelepiped whose edges from $A$ are $\vec{A B}$ , $\vec{A C}$ and $\vec{A D}$ .
Q6Choosing the Right Product

For each geometric question below, state which one of the dot product, the cross product and the scalar triple product settles it, and give one sentence saying why each of the other two cannot. Compute nothing.
1. Two struts leave a joint along $\vec{a}$ and $\vec{b}$ . Is the angle between them acute or obtuse?
2. Three cables leave an anchor along $\vec{a}$ , $\vec{b}$ and $\vec{c}$ . Do all three lie in one plane?
3. A camera arm must point at right angles to both $\vec{a}$ and $\vec{b}$ . Which direction is that?
4. A triangular solar panel has edge vectors $\vec{a}$ and $\vec{b}$ from one corner. How much surface does it present?
Q7Synthesis — drawing on several topics in this set

Let $\vec{u} = (1, - 2, 2)$ and $\vec{v} = (4, 0, - 3)$ .
1. Find the angle between $\vec{u}$ and $\vec{v}$ , to the nearest tenth of a degree.
2. Find ${proj}_{\vec{v}} \vec{u}$ .
3. Find $\vec{u} \times \vec{v}$ and the area of the parallelogram it determines.
4. Verify for these vectors that $‖ \vec{u} \times \vec{v} ‖^{2} = ‖ \vec{u} ‖^{2} ‖ \vec{v} ‖^{2} - (\vec{u} \cdot \vec{v})^{2}$ , and say which of (a)–(c) this identity connects.

The 5 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

Which stream is this for? This is CEGEP Linear Algebra — 201-NYC-05 under the legacy numbering, 201-SN4-RE under the current one, both the same course — written against ministerial competency 0M04, Analyser des problèmes par l'utilisation de concepts de l'algèbre linéaire et de la géométrie vectorielle, in the Sciences de la nature programme. The course's full title names vector geometry alongside linear algebra, and the devis's précisions enumerate this part tightly: exactly three products — the dot product, the cross product and the scalar triple product — plus orthogonal projection. Scope follows the devis rather than any one college's outline, which is why abstract vector spaces and eigenvalues are absent: 0M04 names neither, and what it does ask for is a correct reading of linear independence and dependence in concrete ℝ² and ℝ³. Everything here is in the plane and in space, with real coordinates.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Three products, three different geometric questions

The devis names exactly three, and the reason there are three rather than one is that they answer questions of different shapes. Getting the choice right before you compute anything is most of the work; Q6 is built entirely on that skill and asks you to justify the two rejections as well as the choice, without evaluating a single component.

Ask what kind of answer the question wants, then pick the product that returns it. The dot product takes two vectors and returns a scalar — it settles angles and perpendicularity, and nothing that returns a number can name a direction. The cross product takes two vectors and returns a vector perpendicular to both — it settles directions and areas. The scalar triple product takes three vectors and returns a scalar — it settles volume and coplanarity, and it is the only one of the three that can say anything about three directions at once, because the other two compare vectors two at a time.

u · v = ‖u‖ ‖v‖ cos θ ‖u × v‖ = ‖u‖ ‖v‖ sin θ u · (v × w) = signed volume

Those three lines are the whole map. Q1 works the first, Q3 and Q5 the second, Q4 and Q5 the third — and Q7(d) asks you to verify an identity that ties the first two together, which is the neatest statement in the set of how the dot and cross products divide the work between them.

Choosing a product

Read the question, not the vectors. The first line that matches is the product.

1
Three vectors in the question → scalar triple product
Volume, coplanarity, linear dependence of three directions. A dot or a cross product compares only two vectors at a time, and pairwise facts do not settle a three-vector question — Q6(b) asks you to say exactly that.
2
The answer is a direction → cross product
Anything perpendicular to two given vectors at once. This is where the next worksheet's normals to planes come from.
3
The answer is an area → cross product
‖u × v‖ is the area of the parallelogram on u and v, and half of it is the triangle. Q5 asks for both, one after the other, for exactly this reason.
4
The answer is an angle, or a yes/no about perpendicularity → dot product
The sign alone answers acute or obtuse; zero answers perpendicular. Note that the cross product cannot: sin θ is never negative for θ between 0° and 180°, so ‖u × v‖ takes the same value at θ and at 180° − θ and cannot tell them apart.
5
The answer is a length along a given direction → orthogonal projection
Built from the dot product, and the tool behind every distance formula you will meet in the next set.

The dot product: the sign is the answer

Q1 gives three guy wires leaving one anchor and asks for two dot products, one angle to the nearest tenth of a degree, and then — part (c) — for the geometric statement each number supports. Parts (a) and (b) are arithmetic; part (c) is the question.

Rearranging the defining relation gives the only formula you need for an angle:

cos θ = (u · v) / (‖u‖ ‖v‖), θ = arccos( (u · v) / (‖u‖ ‖v‖) )

Norms are always positive, so the dot product carries the sign of cos θ and nothing else. Positive means the angle is acute, negative means obtuse, zero means the two vectors are orthogonal — and that last line is what "perpendicular" means computationally for the rest of the course, since it needs no angle, no arccosine and no decimal. Part (c) of Q1 asks you to predict a sign from an angle you have already found, which is the same fact read in the other direction.

Two habits keep this clean. Compute ‖u‖ and ‖v‖ before you reach for the arccosine, so that a cosine outside [−1, 1] tells you immediately that something is wrong. And write the conclusion as a sentence about the vectors, not as a number: the marks in part (c) are for "these two wires meet at a right angle", not for the digits that led there.

Orthogonal projection: splitting one vector into two useful pieces

Q2 asks for three things in a deliberate order — the scalar component of u along v, then the projection vector itself, then a decomposition of u into a piece parallel to v and a piece orthogonal to v, with the orthogonality verified.

scalar component = (u · v) / ‖v‖ proj_v u = ( (u · v) / ‖v‖² ) v u = proj_v u + ( u − proj_v u )

One denominator is ‖v‖ and the other is ‖v‖², and mixing them is the standard error here. The distinction is not arbitrary: the first returns a number, a signed length; the second returns a vector, so it needs one more division by ‖v‖ to turn v into a unit vector pointing the right way. Check what kind of object your answer is before you write it down — a projection that is a scalar, or a component that has three coordinates, is the wrong object regardless of the arithmetic.

Notice which vector the subscript names. proj_v u lies along v; it is u that is being taken apart. And the sign of the scalar component tells you whether the projection points the same way as v or the opposite way, which is the dot-product sign rule from Q1 reappearing. Part (c) is the payoff: the leftover piece u − proj_v u is orthogonal to v by construction, and verifying it costs one dot product. Do that check every time — it catches an arithmetic slip in the projection at no cost.

The cross product, and the property that makes it different

Q3 asks you to compute u × v from the 3 × 3 determinant, verify the answer against both inputs, and then write down v × u without computing a second determinant, naming the property you used. That third part is the whole point of the question.

u × v = det [ i j k ; u₁ u₂ u₃ ; v₁ v₂ v₃ ] v × u = −(u × v)

The cross product is anti-commutative: u × v = −(v × u). Unlike the dot product, where u · v and v · u are the same number, the order of the inputs changes the answer — it reverses it. The determinant shows why: swapping the two vectors swaps two rows, and swapping two rows of a determinant reverses its sign. So a cross product taken in the wrong order does not give a wrong number, it gives a vector pointing the wrong way, and a length or an area computed from it looks perfectly correct. Fix an order at the start of a question and keep it.

The middle expansion carries a minus sign on the j term. That is not a convention you can drop; it is what cofactor expansion along the first row produces, and it is the commonest single mistake in the whole set. Part (b) of Q3 is the free check against it: dot your answer with each of the two inputs. A cross product that is not orthogonal to both of its inputs is wrong, and the check takes two lines.

The scalar triple product: three vectors, one number

Q4 gives three vectors and asks for u · (v × w) two ways — as a single 3 × 3 determinant with the three vectors as its rows, and again by computing v × w first and then dotting — before asking, in part (c), what the value tells you about the three vectors.

u · (v × w) = det [ u₁ u₂ u₃ ; v₁ v₂ v₃ ; w₁ w₂ w₃ ] volume = | u · (v × w) |

Zero or non-zero is the entire geometric content. A zero triple product says the parallelepiped built on the three vectors is flat — the three lie in one plane through the origin — which is precisely linear dependence. A non-zero value says they are not coplanar, so they are linearly independent, and its magnitude is the volume they enclose. This is the one product that says something about three directions at once, and it is why Q6(b), a coplanarity question, has only one possible answer.

The two routes in parts (a) and (b) are the same computation — a cofactor expansion along the first row is a dot product with a cross product — which is exactly why the question asks for both. Doing them independently and comparing is the best check available on the sheet.

Areas and volumes: the products doing the measuring

Q5 is four points in space and three measurements: a parallelogram's area, a triangle's area and a parallelepiped's volume. Nothing new is needed — but the first move is one students skip.

Convert points to vectors before anything else. Every one of these formulas eats vectors, not points, so subtract the common corner from the other three and work with the displacement vectors from that corner. Then: the parallelogram's area is ‖AB × AC‖, the triangle is half of it, and the volume is the absolute value of the triple product of the three edge vectors. Take the absolute value: a volume cannot be negative, and the sign of a triple product records only the orientation of the three edges, which the question did not ask about.

Carry the units through. The edges are lengths in metres, so an area is in m² and a volume in m³, and a final answer without them is incomplete on a CEGEP paper.

The synthesis question, and the identity that closes the set

Q7 takes one pair of vectors through the angle, the projection, the cross product and the parallelogram's area — the four techniques of the set, on the same data, so that the numbers can be compared against each other. Then part (d) asks you to verify an identity:

‖u × v‖² = ‖u‖² ‖v‖² − (u · v)²

It is worth reading rather than merely checking. Written with the angle, it is ‖u‖²‖v‖² sin²θ = ‖u‖²‖v‖² (1 − cos²θ), so it is the Pythagorean identity in vector clothing, and it says the cross product measures exactly the part of a pair of vectors that the dot product does not see. At right angles the dot product is zero and the area is as large as it can be; when the vectors are parallel the cross product is the zero vector and the dot product is as large as it can be. Part (d) also asks you to say which of the earlier parts the identity connects, which is the question checking that you noticed.

Where this goes next

Straight into Lines and Planes in Space, the next worksheet in this course, which is built on nothing but these products. A plane is named by a normal vector, and a normal is a cross product. Two lines are parallel, perpendicular or skew according to a dot product and a triple product. Every distance formula in that set — point to line, point to plane, line to line — is an orthogonal projection with the pieces relabelled. If a definition here is still shaky, it will not stay a small gap for long, which is the argument for doing this sheet properly before starting that one.

Getting the most out of it

Decide what kind of answer the question wants first

A number or a direction? Two vectors or three? That single reading fixes the product before any arithmetic starts, and it is the habit Q6 exists to build. A scalar cannot name a direction and a pairwise product cannot settle a three-vector question — those two sentences rule out most wrong choices on their own.

Fix the order and keep it

The dot product does not care about order; the cross product reverses when you swap the inputs. Write the order down at the top of the question, keep it through every part, and when a later part asks for the reversed product, negate rather than recompute. A reversed cross product produces a perfectly plausible vector of the right length pointing the wrong way, so nothing about the answer looks wrong.

Use the free checks — every product on this sheet has one

Dot your cross product with each input and expect zero. Compute a triple product both ways and expect the same number. Dot the leftover piece of a projection with the vector you projected onto and expect zero. Each check costs a line or two and catches the sign slips that this material is full of, particularly the minus on the middle term of a cofactor expansion.

Answer with a sentence, not just a value

Several parts here ask what a number tells you: that two directions are orthogonal, that three are coplanar, that a set is linearly independent. The devis asks for a just interpretation of linear dependence and independence, and on a CEGEP paper that interpretation is looked for in a written clause — "the triple product is non-zero, so the three vectors are not coplanar" — not in the determinant above it.

Want the solutions, or something more challenging?

The worksheet, the questions above and every explanation on this page stay free permanently. Three more PDFs exist for this topic — the worked answer key, a harder problem set, and the answer key to that. They come with the CEGEP Linear Algebra Solutions Bundle, which is what keeps the rest of the series free.

What else exists for Products of Vectors

Three PDFs · 10 pages · all three are in the bundle below.

Answer key — 3 pages. All 7 questions worked step by step, including the restrictions and the justifications. Not a list of final answers.
Challenge problems — 5 pages, 5 problems. A separate sheet at exam-plus difficulty covering the same 6 concepts. Harder than anything on the free sheet.
Challenge answer key — 2 pages. Every challenge problem worked to the same standard, with the checks shown.
PDF, letter size, print-ready.

In the bundle See what's in it Not sold separately

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Common questions

Is this worksheet really free?

Yes — the questions are on this page to read, and the PDF downloads directly, no email and no account. The one paid item is optional: the complete CEGEP Linear Algebra Solutions Bundle, which covers every set at this level.

Is this for 201-NYC-05 or 201-SN4-RE?

Both — they are the same course under two numbering systems, the first the legacy code and the second the current one. The content is written against ministerial competency 0M04 in the Sciences de la nature programme, so it matches whichever code your college prints on the outline.

How do I know which product a question wants?

Ask what kind of answer it is after. An angle, a yes-or-no about perpendicularity, or anything whose answer is a number about two vectors is the dot product. A direction perpendicular to two given vectors, or an area, is the cross product. Anything involving three vectors at once — a volume, or whether they lie in one plane — is the scalar triple product, because the other two only ever compare two vectors at a time.

Why is u × v not the same as v × u?

Because the cross product is anti-commutative: swapping the inputs reverses the result, u × v = −(v × u). The determinant makes it obvious — exchanging the two vectors exchanges two rows, and swapping two rows of a determinant flips its sign. The dot product has no such behaviour, which is why order can be ignored there and never here.

What does it mean when the scalar triple product is zero?

That the three vectors are coplanar: the parallelepiped they build is flat, so it has no volume, and the three directions are linearly dependent. A non-zero value means the opposite — not coplanar, linearly independent — and its absolute value is the volume. It is the one test in this set that looks at three vectors at once.

Do I need determinants before starting this?

A 3 × 3 cofactor expansion, yes — both the cross product and the scalar triple product are determinants in disguise, and the minus sign on the middle term is where most of the errors on this sheet come from. The determinants set in this course covers the mechanics properly if that expansion is not yet automatic.

Is the scalar product the same thing as the dot product?

Yes. The devis says produit scalaire, because the output is a scalar; English usage is usually dot product, and sometimes inner product. The cross product is the produit vectoriel, for the same reason — its output is a vector — and the triple product is the produit mixte.

How is this used in the rest of the course?

Immediately, in Lines and Planes in Space. A plane is specified by a normal vector, which is a cross product; the relative position of two lines is read off a dot product and a triple product; and every distance formula in that set is an orthogonal projection. This sheet is the vocabulary the next one is written in.

Can teachers use this in class?

Yes. Print and photocopy it for your own classes freely — I just ask that the tutorinmontreal.ca footer stays on the page.

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Yes — through one-on-one tutoring, in Montreal or online. Get in touch to arrange a session, or see the current rates.

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