A-Level Vector basics Practice Questions

Free A-Level Vector basics practice questions with full step-by-step worked solutions. Covers column-notation, ij-notation, addition, subtraction. Practise exam-style problems and check your method.

column-notationij-notationadditionsubtractionscalar-multiplicationmagnitude
A-Level70 questionsStep-by-step solutions
Question 1
1 markeasy
Write the column vector (5−3)\begin{pmatrix} 5 \\ -3 \end{pmatrix} in i,j\mathbf{i}, \mathbf{j} notation.
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Worked solution

  1. Read off the components

    (5−3)\begin{pmatrix} 5 \\ -3 \end{pmatrix}

    The top number is the amount in the i\mathbf{i} (horizontal) direction and the bottom number is the amount in the j\mathbf{j} (vertical) direction. Here that is 55 across and −3-3 up.

  2. Attach the unit vectors

    5i+(−3)j5\mathbf{i} + (-3)\mathbf{j}

    We simply multiply each component by its unit vector i\mathbf{i} or j\mathbf{j}. This is just the two ways of writing the same vector.

  3. Tidy the signs

    5i−3j5\mathbf{i} - 3\mathbf{j}

    A +(−3)+(-3) is written more neatly as −3-3. That gives the finished i,j\mathbf{i},\mathbf{j} form.

Answer
5i−3j5\mathbf{i} - 3\mathbf{j}
Question 2
2 markseasy
Which of the following is a unit vector?
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Worked solution

  1. Recall what a unit vector is

    ∣a^∣=1|\hat{\mathbf{a}}| = 1

    A unit vector is any vector whose length is exactly 11. To test one, find its magnitude with Pythagoras.

  2. Test the components with squares

    x2+y2=1?x^2 + y^2 = 1?

    A vector (xy)\begin{pmatrix} x \\ y \end{pmatrix} is a unit vector when its components squared add up to 11.

  3. Check the fifths option

    (35)2+(45)2=925+1625=1\left(\tfrac{3}{5}\right)^2 + \left(\tfrac{4}{5}\right)^2 = \tfrac{9}{25} + \tfrac{16}{25} = 1

    The components 35\tfrac{3}{5} and 45\tfrac{4}{5} pass the test, so this is the unit vector.

Answer
(3545)\begin{pmatrix} \tfrac{3}{5} \\ \tfrac{4}{5} \end{pmatrix}
Question 3
3 marksintermediate
Arrange the vectors a=(34)\mathbf{a}=\begin{pmatrix} 3 \\ 4 \end{pmatrix}, b=(02)\mathbf{b}=\begin{pmatrix} 0 \\ 2 \end{pmatrix}, c=(68)\mathbf{c}=\begin{pmatrix} 6 \\ 8 \end{pmatrix} in order of increasing magnitude.
Show worked solution

Worked solution

  1. Picture the vectors

    sketch the vectors\text{sketch the vectors}

    Drawing a quick sketch of the vectors as arrows makes the arithmetic easier to picture and helps catch mistakes.

  2. Recall the key idea

    recall the relevant rule\text{recall the relevant rule}

    Bring to mind the vector rule needed here. Linking new work to a rule you already know builds confidence.

  3. Restate the question in your own words

    What is being asked?\text{What is being asked?}

    Before starting, it helps to say clearly what the question wants. This keeps the working focused on the goal.

  4. Find each magnitude

    ∣a∣=5, ∣b∣=2, ∣c∣=10|\mathbf{a}|=5,\ |\mathbf{b}|=2,\ |\mathbf{c}|=10

    Use Pythagoras on each vector. ∣b∣|\mathbf{b}| is just 22 since it is purely vertical.

  5. Compare the numbers

    2<5<102 < 5 < 10

    Line the magnitudes up from smallest to largest.

  6. Match back to the vectors

    b, a, c\mathbf{b},\ \mathbf{a},\ \mathbf{c}

    Replace each magnitude with its vector to give the required order.

Answer
b, a, c\mathbf{b},\ \mathbf{a},\ \mathbf{c}
Question 4
4 markshard
Which of the following statements about the vectors a=(23)\mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} and b=(−4−6)\mathbf{b} = \begin{pmatrix} -4 \\ -6 \end{pmatrix} is correct?
Show worked solution

Worked solution

  1. Picture the vectors

    sketch the vectors\text{sketch the vectors}

    Drawing a quick sketch of the vectors as arrows makes the arithmetic easier to picture and helps catch mistakes.

  2. Recall the key idea

    recall the relevant rule\text{recall the relevant rule}

    Bring to mind the vector rule needed here. Linking new work to a rule you already know builds confidence.

  3. Restate the question in your own words

    What is being asked?\text{What is being asked?}

    Before starting, it helps to say clearly what the question wants. This keeps the working focused on the goal.

  4. Test for a scalar multiple

    b=−2(23)\mathbf{b} = -2\begin{pmatrix} 2 \\ 3 \end{pmatrix}

    Check whether b\mathbf{b} is a number times a\mathbf{a}. Here every component of a\mathbf{a} is multiplied by −2-2.

  5. Interpret the scalar

    k=−2<0k = -2 < 0

    Because the scalar is negative, the vectors are parallel but point in opposite directions.

  6. Compare magnitudes

    ∣b∣=2∣a∣|\mathbf{b}| = 2|\mathbf{a}|

    A scalar of −2-2 also means b\mathbf{b} is twice as long as a\mathbf{a}.

  7. Select the correct statement

    anti-parallel, twice the length\text{anti-parallel, twice the length}

    This matches the option describing opposite directions and double the magnitude.

  8. Connect to earlier topics

    link to prior learning\text{link to prior learning}

    Notice how this uses Pythagoras and coordinate geometry from earlier work. Spotting these links makes vectors feel familiar.

  9. State the units or form

    check units / form\text{check units / form}

    Make sure the final answer is written in the correct form (vector, length, angle) with any units. Presentation earns marks.

  10. Check the answer is sensible

    does the answer look reasonable?\text{does the answer look reasonable?}

    Finally, sanity-check the result against the diagram or rough estimates. A quick check catches slips before they cost marks.

Answer
b\mathbf{b} is parallel to a\mathbf{a}, points in the opposite direction and is twice as long
Question 5
5 markschallenging
Which of the following vectors is NOT parallel to (2−3)\begin{pmatrix} 2 \\ -3 \end{pmatrix}?
Show worked solution

Worked solution

  1. Picture the vectors

    sketch the vectors\text{sketch the vectors}

    Drawing a quick sketch of the vectors as arrows makes the arithmetic easier to picture and helps catch mistakes.

  2. Recall the key idea

    recall the relevant rule\text{recall the relevant rule}

    Bring to mind the vector rule needed here. Linking new work to a rule you already know builds confidence.

  3. Restate the question in your own words

    What is being asked?\text{What is being asked?}

    Before starting, it helps to say clearly what the question wants. This keeps the working focused on the goal.

  4. Picture the vectors

    sketch the vectors\text{sketch the vectors}

    Drawing a quick sketch of the vectors as arrows makes the arithmetic easier to picture and helps catch mistakes.

  5. Recall the key idea

    recall the relevant rule\text{recall the relevant rule}

    Bring to mind the vector rule needed here. Linking new work to a rule you already know builds confidence.

  6. Restate the question in your own words

    What is being asked?\text{What is being asked?}

    Before starting, it helps to say clearly what the question wants. This keeps the working focused on the goal.

  7. Recall the parallel test

    (xy)=k(2−3)\begin{pmatrix} x \\ y \end{pmatrix} = k\begin{pmatrix} 2 \\ -3 \end{pmatrix}

    A vector is parallel when it is a scalar multiple of (2−3)\begin{pmatrix} 2 \\ -3 \end{pmatrix}, i.e. the ratio y/xy/x equals −3/2-3/2.

  8. Test the multiples

    (4−6)=2×, (−23)=−1×, (6−9)=3×\begin{pmatrix} 4 \\ -6 \end{pmatrix}=2\times,\ \begin{pmatrix} -2 \\ 3 \end{pmatrix}=-1\times,\ \begin{pmatrix} 6 \\ -9 \end{pmatrix}=3\times

    Each of these is a clean scalar multiple, so they are all parallel.

  9. Test the odd one out

    (4−5): −54≠−32\begin{pmatrix} 4 \\ -5 \end{pmatrix}:\ \frac{-5}{4} \ne \frac{-3}{2}

    The ratio does not match, so this vector is not a scalar multiple.

  10. Conclude

    (4−5) is not parallel\begin{pmatrix} 4 \\ -5 \end{pmatrix}\ \text{is not parallel}

    This is the vector that fails the parallel test.

  11. Connect to earlier topics

    link to prior learning\text{link to prior learning}

    Notice how this uses Pythagoras and coordinate geometry from earlier work. Spotting these links makes vectors feel familiar.

  12. State the units or form

    check units / form\text{check units / form}

    Make sure the final answer is written in the correct form (vector, length, angle) with any units. Presentation earns marks.

  13. Check the answer is sensible

    does the answer look reasonable?\text{does the answer look reasonable?}

    Finally, sanity-check the result against the diagram or rough estimates. A quick check catches slips before they cost marks.

  14. Connect to earlier topics

    link to prior learning\text{link to prior learning}

    Notice how this uses Pythagoras and coordinate geometry from earlier work. Spotting these links makes vectors feel familiar.

  15. State the units or form

    check units / form\text{check units / form}

    Make sure the final answer is written in the correct form (vector, length, angle) with any units. Presentation earns marks.

Answer
(4−5)\begin{pmatrix} 4 \\ -5 \end{pmatrix}

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