Secondary 5 Vectors and Matrices Worksheet

All 16 questions

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

1. Q1Adding and Subtracting Vectors

   Let $\overrightarrow{u}=(5,-2)$ and $\overrightarrow{v}=(-3,6)$.

   1. Find $\overrightarrow{u}+\overrightarrow{v}$ and $\overrightarrow{u}-\overrightarrow{v}$.
   2. Give the exact value of $\Vert \overrightarrow{u}+\overrightarrow{v}\Vert$, then a decimal value to the nearest hundredth.
   3. Using the Chasles relation, simplify $\overrightarrow{RS}+\overrightarrow{TR}-\overrightarrow{US}$ into a single vector.
2. Q2Comparing Two Vectors

   Consider $\overrightarrow{a}=(4,-6)$, $\overrightarrow{b}=(-6,-4)$, $\overrightarrow{c}=(-2,3)$, $\overrightarrow{d}=(-4,6)$ and $\overrightarrow{e}=(6,-9)$. Support every answer with a calculation.

   1. Which vector is the opposite of $\overrightarrow{a}$?
   2. Which vectors are collinear with $\overrightarrow{a}$ without being equal or opposite to it?
   3. Which vector is orthogonal to $\overrightarrow{a}$?
   4. For $P(1,1)$ and $Q(5,-5)$, is $\overrightarrow{PQ}$ equal to $\overrightarrow{a}$?
3. Q3Linear Algebra

   For each expression below, state whether the result is a scalar, a vector, a matrix (give its dimension) or is not defined, and say why. Here $\overrightarrow{u}$ and $\overrightarrow{v}$ are vectors of the plane, $A$ is a $2\times 3$ matrix, $B$ is a $2\times 3$ matrix and $C$ is a $3\times 4$ matrix.

   1. $\overrightarrow{u}·\overrightarrow{v}$
   2. $\Vert \overrightarrow{u}\Vert \overrightarrow{v}$
   3. $\overrightarrow{u}+\Vert \overrightarrow{v}\Vert$
   4. $AB$
   5. $AC$
4. Q4Linear Combination of Vectors

   Let $\overrightarrow{u}=(2,1)$ and $\overrightarrow{v}=(-1,3)$.

   1. Write $\overrightarrow{w}=(8,-3)$ as a linear combination $a\overrightarrow{u}+b\overrightarrow{v}$.
   2. Do the same for $\overrightarrow{z}=(-6,11)$.
   3. Explain why every vector of the plane can be written as a linear combination of $\overrightarrow{u}$ and $\overrightarrow{v}$.
5. Q5Matrices

   Let $$A=\left(\begin{array}{ccc}3& -1& 5\\ 0& 2& -4\end{array}\right),B=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],D=\left[\begin{array}{c}-2\\ 7\\ 0\end{array}\right].$$

   1. Give the dimension of $A$ and the value of the element $a_{23}$.
   2. Write $A^T$ and give its dimension.
   3. Name the type of each of $B$ and $D$.
6. Q6Matrix Operations

   Let $A=\left(\begin{array}{cc}2& -1\\ 0& 3\end{array}\right)$ and $B=\left(\begin{array}{cc}4& 1\\ -2& 5\end{array}\right)$. Compute:

   1. $2A-B$
   2. $AB$
7. Q7Multiplying Vectors by a Scalar and the Scalar Product (Dot Product)

   Let $\overrightarrow{u}=(6,-3)$ and $\overrightarrow{v}=(-2,5)$.

   1. Compute $2\overrightarrow{u}-3\overrightarrow{v}$.
   2. Compute $\overrightarrow{u}·\overrightarrow{v}$.
   3. Find the angle between $\overrightarrow{u}$ and $\overrightarrow{v}$, to the nearest degree.
8. Q8Proving Vector Statements

   In a triangle $ABC$, let $M$ be the midpoint of $\overline{AB}$ and $N$ the midpoint of $\overline{AC}$.

   1. Using only the Chasles relation and the properties of vector operations, prove that $\overrightarrow{MN}=\frac{1}{2}\overrightarrow{BC}$.
   2. State the two geometric facts about the segments $\overline{MN}$ and $\overline{BC}$ that follow from this vector equality.
9. Q9Solving First- or Second-Degree System of Equations (Semi-Linear)

   Solve the system algebraically and interpret the result geometrically. $$\{\begin{array}{c}y=2x+1\hfill \\ [\mathrm{0.2em}]y=x^2-3x+7\hfill \end{array}$$

10. Q10Solving Problems Involving Vectors

    Two tugboats pull a barge. Tug $A$ pulls with a force of $4000$ N directed ${40}^{\circ }$ above the eastward axis; tug $B$ pulls with $3000$ N directed ${25}^{\circ }$ below that same axis.

    1. Write each force in components (nearest newton).
    2. Find the norm of the resultant force, to the nearest newton, and its direction relative to the eastward axis, to the nearest tenth of a degree.
11. Q11The Orthogonal Projection of a Vector

    Let $\overrightarrow{u}=(7,1)$ and $\overrightarrow{v}=(3,4)$.

    1. Find the orthogonal projection of $\overrightarrow{u}$ onto $\overrightarrow{v}$ and give its norm.
    2. Compute $\overrightarrow{u}-{\overrightarrow{u}}_{\overrightarrow{v}}$ and verify that it is orthogonal to $\overrightarrow{v}$.
12. Q12The Properties of Vector Operations

    For each statement, decide whether it is true for all vectors $\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$ of the plane and all $k\in ℝ$. If it is true, name the property; if it is false, give an explicit counterexample.

    1. $\Vert \overrightarrow{u}+\overrightarrow{v}\Vert =\Vert \overrightarrow{u}\Vert +\Vert \overrightarrow{v}\Vert$
    2. $\Vert k\overrightarrow{u}\Vert =k\Vert \overrightarrow{u}\Vert$
    3. $\overrightarrow{u}·\overrightarrow{v}=\overrightarrow{v}·\overrightarrow{u}$
    4. $(\overrightarrow{u}·\overrightarrow{v})\overrightarrow{w}=\overrightarrow{u}(\overrightarrow{v}·\overrightarrow{w})$
13. Q13Transformation Matrices

    Points of the plane are written as column matrices $\left[\begin{array}{c}x\\ y\end{array}\right]$ and transformed by multiplying on the left by a $2\times 2$ matrix. Consider the triangle $A(2,1)$, $B(5,1)$, $C(5,3)$.

    1. Write the matrix $R$ of the rotation of ${90}^{\circ }$ counterclockwise about the origin, and find the images $A^{\prime }$, $B^{\prime }$, $C^{\prime }$.
    2. Write the matrix $H$ of the dilation centred at the origin with ratio $3$, and find the image of $A$.
14. Q14Vector Components
    1. A vector $\overrightarrow{u}$ has norm $12$ and its direction makes an angle of ${210}^{\circ }$ with the positive $x$-axis. Give its components, exactly and then to the nearest hundredth.
    2. Give the norm of $\overrightarrow{v}=(-5,12)$ and its direction, to the nearest tenth of a degree.
15. Q15Vectors

    Let $P(-3,2)$ and $Q(5,-4)$.

    1. Give the components of $\overrightarrow{PQ}$ and its norm.
    2. Give the unit vector having the same direction and sense as $\overrightarrow{PQ}$.
    3. Find the coordinates of the point $R$ such that $\overrightarrow{QR}=\overrightarrow{PQ}$.
16. Q16Synthesis — drawing on several sheets in this topic

    A logo is built from the triangle $O(0,0)$, $P(4,0)$, $Q(4,3)$. It is first rotated ${90}^{\circ }$ counterclockwise about the origin, then enlarged by a dilation centred at the origin with ratio $2$.

    1. Write the single matrix $M$ that performs both transformations, and state whether reversing their order would change $M$.
    2. Find the image $Q^{\prime }$ of $Q$.
    3. Using the dot product and norms, verify that $\Vert \overrightarrow{O{Q}^{\prime }}\Vert =2\Vert \overrightarrow{OQ}\Vert$ and that $\overrightarrow{OQ}⟂\overrightarrow{O{Q}^{\prime }}$.

The 13 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

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.

A vector is a displacement, not a place (Q15, Q14)

Everything on this sheet is easier once that sentence is settled. The components of the vector from P to Q are the arrival minus the departure:

Its norm is Pythagoras on those components, and a unit vector with the same direction and sense is the vector divided by its own norm — always check afterwards that the components of your unit vector square-and-add to 1. Q15(c) then runs the idea backwards: knowing a vector and a starting point, the endpoint is the point plus the components.

Going the other way, components from a norm and a direction, is (‖u‖ cos θ, ‖u‖ sin θ) — and the signs the formula produces are themselves the check. A direction of 210° is in the third quadrant, so both components must come out negative. If one of yours is positive, you have made an arithmetic slip, not discovered an exception.

The Chasles relation: make the heads meet the tails (Q1, Q8)

Chasles says a trip from A to C is the same displacement as the trip A to B followed by B to C, for any intermediate point B — including one that is nowhere near the segment. Reading it left to right joins two vectors; reading it right to left inserts a point of your choosing, and that is the move most proofs need.

Q8 also asks what the vector equality tells you geometrically, and that is a separate mark. A vector being a scalar multiple of another gives parallel; taking norms on both sides gives the length ratio. Two facts, two sentences.

Equal, opposite, collinear, orthogonal (Q2)

Q2 asks for four different relations between vectors, each "supported by a calculation" — an assertion with no arithmetic under it earns nothing. The tests:

Collinear — one is a scalar multiple of the other. In practice, check whether u₁v₂ − u₂v₁ = 0; if it is, divide a pair of components to find the factor. Opposite — collinear with factor exactly −1. Equal (equipollent) — identical components, whatever the position of the arrow on the page. Orthogonal — the scalar product is 0.

Linear combinations, and why two vectors are enough (Q4)

Writing w = a·u + b·v in components gives two equations in the two unknowns a and b. Solve by substitution and always substitute back — the check is one line and catches every sign error.

Part (c) asks why every vector of the plane can be reached, and an example is not an answer. The argument: u and v are not collinear, which means the system built from their components has a non-zero determinant, which means it has exactly one solution for any right-hand side. Two non-collinear vectors therefore form a basis, and the coefficients are unique. Had they been collinear, every combination would have stayed on a single line through the origin — most vectors unreachable, and the reachable ones reachable in infinitely many ways.

The scalar product tells you the angle before you compute it (Q7)

The scalar product of two vectors in components is u₁v₁ + u₂v₂ — a number, never a vector. From it:

Before touching the inverse cosine, read the sign. Positive means an acute angle, zero means a right angle, negative means an obtuse one — so a negative scalar product with an answer of 40° is wrong on sight. Keep the norms as exact square roots until the very last step; rounding two norms to two decimals each and then dividing is how an angle drifts by a degree.

Orthogonal projection: the formula divides by the norm squared (Q11)

The projection of u onto v is the part of u that lies along v — a vector, parallel to v, obtained by scaling v. That is why the denominator is ‖v‖² and not ‖v‖: one factor of ‖v‖ converts the scalar product into a length along v, and the second turns v itself into a direction of length 1. Writing ‖v‖ once instead of twice is the standard error, and it produces a vector pointing the right way with the wrong length.

Matrices: the dimensions decide what is even possible (Q3, Q5, Q6)

Q3 asks nothing but "what kind of object is this, and is it defined?", which is a fair description of half the marks on a matrix exam.

Two conventions that get confused constantly. The dimension is written rows × columns, and the element a₂₃ is row 2, column 3 — row index first, always. The transpose exchanges rows and columns, so an m × n matrix transposes to an n × m one. For the product itself (Q6), each entry is a row of the left matrix run against a column of the right one: entry in row i, column j comes from row i of the left and column j of the right.

Transformation matrices: build them, don't memorise them (Q13, Q16)

A 2 × 2 matrix acting on column matrices is completely determined by what it does to the two basic directions, and those two images are literally its columns. That is the whole trick: to write the matrix of a quarter-turn counterclockwise, ask where (1, 0) goes — to (0, 1) — and where (0, 1) goes — to (−1, 0) — and write those two answers down as the first and second columns. No memorisation, and it works for reflections and dilations equally.

A dilation centred at the origin with ratio k is k times the identity, which is why it multiplies every length by k and changes no angle. In a composition, the matrix applied first is written on the right, next to the point it acts on — so "rotate, then enlarge" is the dilation times the rotation. Q16 asks you to build the single matrix doing both, then to verify with norms and a scalar product that the image is twice as long and perpendicular to the original. That verification is the point of the question: it is the transformation's geometric meaning, recovered from arithmetic.

True, false, and what counts as proof (Q12)

Q12 gives four claims about vector operations and asks you to name the property or produce a counterexample. The asymmetry is the thing to internalise: a false claim needs one explicit counterexample — actual vectors, actual arithmetic, both sides evaluated and seen to differ — while a true claim needs a general argument, since no number of examples proves it.

Two of the four are worth knowing in their corrected form. The norm of a sum is not the sum of the norms; it is at most that, with equality only when the two vectors have the same sense — the triangle inequality, and geometrically obvious the moment you draw two arrows head to tail. And scaling a vector multiplies its norm by |k|, not by k, because a norm is never negative.

A line meeting a parabola (Q9)

Substituting the linear equation into the quadratic one collapses the system to a single second-degree equation. Its number of real roots is the geometry: two roots means the line cuts the curve twice, one (double) root means it is tangent, no real root means they never meet. Q9 ends by asking for that interpretation, so finish with the sentence, not with the pair of coordinates — and substitute both solutions back into the parabola, since a slip there is invisible if you only check the line.

Forces, and why components come first (Q10)

Two forces pulling at different angles cannot be added as numbers. Resolve each into (magnitude · cos θ, magnitude · sin θ), add the components separately, then rebuild the resultant's norm with Pythagoras and its direction with an inverse tangent placed by quadrant. An angle measured below the reference axis is negative — feed it in as a negative angle and the sine will take care of the sign of the vertical component for you.

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Getting the most out of it

Draw the axes, even for a two-line question

A sketch with the two components marked settles the quadrant before you touch an inverse trigonometric function, and it makes an impossible answer look wrong instead of looking finished. On the projection and angle questions it also tells you in advance whether to expect a negative scalar product.

Name the object before you compute it

Say "this will be a number" or "this will be a 2 × 4 matrix" out loud before starting. It takes a second, it is exactly what one of the questions asks for in its own right, and it stops you from writing an answer whose shape cannot possibly be right.

Substitute back, every time

Every question on this sheet has a cheap check built into it: the coefficients of a linear combination can be recomputed, a unit vector's components square-and-add to 1, the leftover of a projection has scalar product 0 with the direction, the solutions of the system satisfy the parabola. Doing the check is a habit worth more than any individual technique here.

Common questions