Vector notation and magnitude questions with answers

Questions

1. Question 1A vector as a column vector

   The diagram shows the vector $\overrightarrow{RS}$ on a grid of unit squares. Express $\overrightarrow{RS}$ as a column vector.

   Show the worked solution

   1. Step 1: Count across from R to S (right is +, left is −)$x\text{-component} = 5$
   2. Step 2: Count up or down (up is +, down is −)$y\text{-component} = 3$
   3. Step 3: Write it as a column vector$\overrightarrow{RS} = \begin{pmatrix} 5 \\ 3 \end{pmatrix}$
2. Question 2A vector as a column vector

   The diagram shows the vector $\overrightarrow{AB}$ on a grid of unit squares. Express $\overrightarrow{AB}$ as a column vector.

   Show the worked solution

   1. Step 1: Count across from A to B (right is +, left is −)$x\text{-component} = 3$
   2. Step 2: Count up or down (up is +, down is −)$y\text{-component} = 2$
   3. Step 3: Write it as a column vector$\overrightarrow{AB} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$
3. Question 3Magnitude

   $\mathbf{b} = \begin{pmatrix} 12 \\ 5 \end{pmatrix}$. Find $|\mathbf{b}|$.

   Show the worked solution

   1. Step 1: Square and add the components$12^2 + 5^2 = 169$
   2. Step 2: Take the square root$|\mathbf{b}| = \sqrt{169}$$|\mathbf{b}| = 13$
4. Question 4Magnitude

   $\mathbf{q} = \begin{pmatrix} -7 \\ 1 \end{pmatrix}$. Find $|\mathbf{q}|$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Square and add the components$(-7)^2 + 1^2 = 50$
   2. Step 2: Take the square root$|\mathbf{q}| = \sqrt{50}$$|\mathbf{q}| = 7.07$ (3 s.f.)
5. Question 5An unknown from the magnitude

   $\mathbf{p} = \begin{pmatrix} -12 \\ k \end{pmatrix}$ and $|\mathbf{p}| = 15$. Find the value of $k$, given that $k > 0$.

   Show the worked solution

   1. Step 1: Magnitude: square and add the components$k^2 + (-12)^2 = 15^2$$k^2 = 81$
   2. Step 2: Take the positive square root$k = 9$
6. Question 6An unknown from the magnitude

   $\mathbf{p} = \begin{pmatrix} 12 \\ k \end{pmatrix}$ and $|\mathbf{p}| = 20$. Find the value of $k$, given that $k > 0$.

   Show the worked solution

   1. Step 1: Magnitude: square and add the components$k^2 + 12^2 = 20^2$$k^2 = 256$
   2. Step 2: Take the positive square root$k = 16$