Pure Mathematics Practice: Vectors

Original practice material

These four structured questions total 27 marks. They practise vector products, lines, planes and linked spatial reasoning suitable for Paper 1 or Paper 2 Section A.

Answers: Vectors Answers.

Question 1: Scalar and Vector Products

Let

$$\mathbf{a}=\left(\begin{array}{c}2\\ -1\\ 3\end{array}\right),\mathbf{b}=\left(\begin{array}{c}1\\ 4\\ -2\end{array}\right).$$

(a) Find $\mathbf{a}\cdot \mathbf{b}$. [2]

(b) Find $\mathbf{a}\times \mathbf{b}$. [3]

(c) Find the angle between $\mathbf{a}$ and $\mathbf{b}$ to the nearest degree. [3]

[Total: 8]

Question 2: Line Through Two Points

Points $A(1,2,-1)$ and $B(4,0,5)$ lie on a line $l$.

(a) Write a vector equation for $l$. [3]

(b) Determine whether $P(7,-2,11)$ lies on $l$, showing how the line parameter is checked in all three coordinates. [3]

[Total: 6]

Question 3: Plane Equation and Distance

The plane $\Pi$ passes through $A(1,0,2)$ and has normal vector

$$\mathbf{n}=\left(\begin{array}{c}2\\ -1\\ 3\end{array}\right).$$

(a) Find a Cartesian equation of $\Pi$. [3]

(b) Find the perpendicular distance from $P(3,4,1)$ to $\Pi$. [4]

[Total: 7]

Question 4: Line and Plane

The line $l$ has equation

$$\mathbf{r}=\left(\begin{array}{c}1\\ 2\\ 0\end{array}\right)+\lambda \left(\begin{array}{c}2\\ -1\\ 1\end{array}\right),$$

and the plane $\Pi$ has equation $x+y+z=8$.

(a) Find the point where $l$ meets $\Pi$. [3]

(b) Find the acute angle between $l$ and $\Pi$, giving your answer to 1 decimal place. [3]

[Total: 6]