Vector geometry Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Vector geometry questions. See exactly how to solve problems on vector-addition, components, vector-subtraction, column-vectors.

vector-additioncomponentsvector-subtractioncolumn-vectorsscalar-multiplelinear-combination
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Given a=2i+5j\mathbf{a}=2\mathbf{i} + 5\mathbf{j} and b=4i−j\mathbf{b}=4\mathbf{i} - \mathbf{j}, find a+b\mathbf{a}+\mathbf{b}.

Worked solution

  1. Write each vector in component form

    a=2i+5j,b=4i−j\mathbf{a}=2\mathbf{i} + 5\mathbf{j},\quad \mathbf{b}=4\mathbf{i} - \mathbf{j}

    We line the vectors up by their i\mathbf{i} (across) and j\mathbf{j} (up) parts so we can add matching parts together.

  2. Add the i\mathbf{i} components and the j\mathbf{j} components

    (2+4)i+(5+(−1))j(2+4)\mathbf{i}+(5+(-1))\mathbf{j}

    Adding vectors just means adding the across-parts together and the up-parts together separately.

  3. Simplify

    6i+4j6\mathbf{i}+4\mathbf{j}

    This single vector is the result of the addition.

Answer
6i+4j6\mathbf{i}+4\mathbf{j}
Question 2
2 markseasy
Given a=(72)\mathbf{a}=\begin{pmatrix} 7 \\ 2 \end{pmatrix} and b=(35)\mathbf{b}=\begin{pmatrix} 3 \\ 5 \end{pmatrix}, find a−b\mathbf{a}-\mathbf{b}.

Worked solution

  1. Write the vectors as columns

    a=(72),b=(35)\mathbf{a}=\begin{pmatrix} 7 \\ 2 \end{pmatrix},\quad \mathbf{b}=\begin{pmatrix} 3 \\ 5 \end{pmatrix}

    A column vector stacks the across-part on top of the up-part.

  2. Subtract component by component

    (4−3)\begin{pmatrix} 4 \\ -3 \end{pmatrix}

    Subtracting a vector means subtracting the top numbers and the bottom numbers separately.

  3. State the answer

    (4−3)\begin{pmatrix} 4 \\ -3 \end{pmatrix}

    The negative bottom number just means the result points downwards.

Answer
(4−3)\begin{pmatrix} 4 \\ -3 \end{pmatrix}
Question 3
2 markseasy
Given a=3i−2j\mathbf{a}=3\mathbf{i} - 2\mathbf{j}, find 4a4\mathbf{a}.

Worked solution

  1. Write the vector

    a=3i−2j\mathbf{a}=3\mathbf{i} - 2\mathbf{j}

    We are scaling this vector to make it four times as long in the same line of direction.

  2. Multiply each component by 4

    4×3 i+4×(−2) j4\times 3\,\mathbf{i}+4\times(-2)\,\mathbf{j}

    Multiplying by a number (a scalar) stretches the vector, so every component is multiplied by that number.

  3. Simplify

    12i−8j12\mathbf{i}-8\mathbf{j}

    This vector points the same way as a\mathbf{a} but is four times longer.

Answer
12i−8j12\mathbf{i}-8\mathbf{j}
Question 4
3 markseasy
Given a=i+3j\mathbf{a}=\mathbf{i} + 3\mathbf{j} and b=4i−2j\mathbf{b}=4\mathbf{i} - 2\mathbf{j}, find 2a+b2\mathbf{a}+\mathbf{b}.

Worked solution

  1. Work out 2a2\mathbf{a} first

    2a=2(i+3j)=2i+6j2\mathbf{a}=2(\mathbf{i} + 3\mathbf{j})=2\mathbf{i} + 6\mathbf{j}

    Deal with the multiplication before the addition, exactly like normal order of operations.

  2. Write b\mathbf{b}

    b=4i−2j\mathbf{b}=4\mathbf{i} - 2\mathbf{j}

    Now we have two vectors ready to add together.

  3. Add the two vectors

    (2+4)i+(6+(−2))j(2+4)\mathbf{i}+(6+(-2))\mathbf{j}

    Add the across-parts and the up-parts separately.

  4. Simplify

    6i+4j6\mathbf{i}+4\mathbf{j}

    This is the combined vector.

Answer
6i+4j6\mathbf{i}+4\mathbf{j}
Question 5
3 markseasy
Given a=2i+j\mathbf{a}=2\mathbf{i} + \mathbf{j} and b=i−3j\mathbf{b}=\mathbf{i} - 3\mathbf{j}, find 3a−2b3\mathbf{a}-2\mathbf{b}.

Worked solution

  1. Work out 3a3\mathbf{a}

    3a=6i+3j3\mathbf{a}=6\mathbf{i} + 3\mathbf{j}

    Multiply both components of a\mathbf{a} by 3.

  2. Work out 2b2\mathbf{b}

    2b=2i−6j2\mathbf{b}=2\mathbf{i} - 6\mathbf{j}

    Multiply both components of b\mathbf{b} by 2.

  3. Subtract

    (6−2)i+(3−(−6))j(6-2)\mathbf{i}+(3-(-6))\mathbf{j}

    Subtract the second vector from the first, part by part. Take care with the double negative.

  4. Simplify

    4i+9j4\mathbf{i}+9\mathbf{j}

    This is the final combined vector.

Answer
4i+9j4\mathbf{i}+9\mathbf{j}

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