Topics › KS4 › Geometry & measures · Foundation & Higher

Vectors

1. vec-01 Present from here

   $\mathbf{a} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$. What is $\mathbf{a} + \mathbf{b}$?

   A Option A: $\begin{pmatrix} 4 \\ -7 \end{pmatrix}$

   B Option B: $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$

   C Option C: $\begin{pmatrix} 8 \\ -3 \end{pmatrix}$

   D Option D: $\begin{pmatrix} -3 \\ -10 \end{pmatrix}$
   Answer and misconceptions

   A Option A: $\begin{pmatrix} 4 \\ -7 \end{pmatrix}$ ✗

   Subtracts the vectors instead of addingWorks out $3 - (-1) = 4$ and $-2 - 5 = -7$, which is $\mathbf{a} - \mathbf{b}$.

   B Option B: $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ ✓ Correct

   C Option C: $\begin{pmatrix} 8 \\ -3 \end{pmatrix}$ ✗

   Adds the top of one vector to the bottom of the otherWorks out $3 + 5$ and $-2 + (-1)$; add top to top and bottom to bottom.

   D Option D: $\begin{pmatrix} -3 \\ -10 \end{pmatrix}$ ✗

   Multiplies the components instead of addingWorks out $3 \times (-1)$ and $-2 \times 5$; adding vectors means adding matching components.
2. vec-02 Present from here

   $\mathbf{a} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}$. What is $3\mathbf{a}$?

   A Option A: $\begin{pmatrix} 7 \\ 2 \end{pmatrix}$

   B Option B: $\begin{pmatrix} 12 \\ -1 \end{pmatrix}$

   C Option C: $\begin{pmatrix} 12 \\ 3 \end{pmatrix}$

   D Option D: $\begin{pmatrix} 12 \\ -3 \end{pmatrix}$
   Answer and misconceptions

   A Option A: $\begin{pmatrix} 7 \\ 2 \end{pmatrix}$ ✗

   Adds the scalar instead of multiplyingWorks out $4 + 3$ and $-1 + 3$; $3\mathbf{a}$ means $3 \times$ each component.

   B Option B: $\begin{pmatrix} 12 \\ -1 \end{pmatrix}$ ✗

   Multiplies only the top componentMultiplies $4$ by $3$ but leaves $-1$; both components are multiplied by $3$.

   C Option C: $\begin{pmatrix} 12 \\ 3 \end{pmatrix}$ ✗

   Thinks multiplying a negative gives a positiveWorks out $3 \times (-1) = 3$; a positive times a negative is negative, $-3$.

   D Option D: $\begin{pmatrix} 12 \\ -3 \end{pmatrix}$ ✓ Correct
3. vec-03 Present from here

   $A$ is the point $(2, 5)$ and $B$ is the point $(7, 1)$. What is the column vector $\overrightarrow{AB}$?

   A Option A: $\begin{pmatrix} -5 \\ 4 \end{pmatrix}$

   B Option B: $\begin{pmatrix} 9 \\ 6 \end{pmatrix}$

   C Option C: $\begin{pmatrix} 5 \\ -4 \end{pmatrix}$

   D Option D: $\begin{pmatrix} 7 \\ 1 \end{pmatrix}$
   Answer and misconceptions

   A Option A: $\begin{pmatrix} -5 \\ 4 \end{pmatrix}$ ✗

   Subtracts the end point from the start pointWorks out $A - B$, which is $\overrightarrow{BA}$; $\overrightarrow{AB}$ is $B - A$: $7 - 2 = 5$ and $1 - 5 = -4$.

   B Option B: $\begin{pmatrix} 9 \\ 6 \end{pmatrix}$ ✗

   Adds the coordinatesWorks out $2 + 7$ and $5 + 1$; the vector is the change from $A$ to $B$, so subtract.

   C Option C: $\begin{pmatrix} 5 \\ -4 \end{pmatrix}$ ✓ Correct

   D Option D: $\begin{pmatrix} 7 \\ 1 \end{pmatrix}$ ✗

   Gives the coordinates of the end pointWrites $B$ as a column vector; that is the journey from the origin to $B$, not from $A$ to $B$.
4. vec-04 Present from here

   $\mathbf{a} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$. What is $\mathbf{a} - 2\mathbf{b}$?

   A Option A: $\begin{pmatrix} -1 \\ 4 \end{pmatrix}$

   B Option B: $\begin{pmatrix} 11 \\ 0 \end{pmatrix}$

   C Option C: $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$

   D Option D: $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$
   Answer and misconceptions

   A Option A: $\begin{pmatrix} -1 \\ 4 \end{pmatrix}$ ✓ Correct

   B Option B: $\begin{pmatrix} 11 \\ 0 \end{pmatrix}$ ✗

   Adds instead of subtractingWorks out $\mathbf{a} + 2\mathbf{b}$: $5 + 6 = 11$ and $2 + (-2) = 0$.

   C Option C: $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ ✗

   Ignores the scalar 2Works out $\mathbf{a} - \mathbf{b}$; $\mathbf{b}$ must be doubled to $\begin{pmatrix} 6 \\ -2 \end{pmatrix}$ first.

   D Option D: $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$ ✗

   Treats subtracting a negative as subtractingWorks out $2 - (-2) = 0$; subtracting a negative is adding, so $2 - (-2) = 4$.
5. vec-05 Present from here

   $\overrightarrow{AB} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$. What is $\overrightarrow{BA}$?

   A Option A: $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$

   B Option B: $\begin{pmatrix} -3 \\ -2 \end{pmatrix}$

   C Option C: $\begin{pmatrix} -2 \\ 3 \end{pmatrix}$

   D Option D: $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$
   Answer and misconceptions

   A Option A: $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ ✗

   Thinks direction does not matterGives the same vector; going from $B$ to $A$ is the opposite journey, so both signs change.

   B Option B: $\begin{pmatrix} -3 \\ -2 \end{pmatrix}$ ✗

   Makes every component negativePuts a minus sign on both numbers; $-(-2) = 2$, so the bottom becomes positive.

   C Option C: $\begin{pmatrix} -2 \\ 3 \end{pmatrix}$ ✗

   Reverses the order of the componentsSwaps the top and bottom numbers; reversing the direction keeps the order and changes each sign.

   D Option D: $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$ ✓ Correct