Scalar multiples and parallel vectors questions with answers

Questions

1. Question 1A multiple of a vector

   $\mathbf{a} = \begin{pmatrix} -5 \\ -2 \end{pmatrix}$. Find $4\mathbf{a}$.

   Show the worked solution

   1. Step 1: Multiply each component$4 \times (-5) = -20$$4 \times (-2) = -8$
   2. Step 2: Write the column vector$4\mathbf{a} = \begin{pmatrix} -20 \\ -8 \end{pmatrix}$
2. Question 2Combining multiples

   $\mathbf{a} = \begin{pmatrix} -3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ -5 \end{pmatrix}$. Find $\mathbf{a} - 3\mathbf{b}$.

   Show the worked solution

   1. Step 1: Multiply, then combine, top numbers$1 \times (-3) + (-3) \times (-1) = 0$
   2. Step 2: Then the bottom numbers$1 \times 1 + (-3) \times (-5) = 16$
   3. Step 3: Write the column vector$\mathbf{a} - 3\mathbf{b} = \begin{pmatrix} 0 \\ 16 \end{pmatrix}$
3. Question 3Parallel vectors

   $\mathbf{p} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} -6 \\ h \end{pmatrix}$. Given that $\mathbf{p}$ and $\mathbf{q}$ are parallel, find the value of $h$.

   Show the worked solution

   1. Step 1: Parallel vectors: how are p and q related?$\mathbf{q} = m\mathbf{p}$
   2. Step 2: Use the top numbers to find m$2 m = -6$$m = -3$
   3. Step 3: Then the bottom numbers$h = -3 \times 3$$h = -9$
4. Question 4Magnitude of a combination

   $\mathbf{a} = \begin{pmatrix} -3 \\ -4 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ -5 \end{pmatrix}$. Find $|2\mathbf{a} + \mathbf{b}|$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Multiply, then combine, top numbers$2 \times (-3) + 1 \times (-1) = -7$
   2. Step 2: Then the bottom numbers$2 \times (-4) + 1 \times (-5) = -13$
   3. Step 3: Magnitude: square, add, square root$|2\mathbf{a} + \mathbf{b}| = \sqrt{(-7)^2 + (-13)^2}$$|2\mathbf{a} + \mathbf{b}| = \sqrt{218}$$|2\mathbf{a} + \mathbf{b}| = 14.8$ (3 s.f.)