IB Mathematics Analysis and Approaches Topic AHL…

Consider points $A(1,-3,4)$ , $B(3,-2,6)$ , $C(2,-1,2)$ .

(a) Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .

(b) Compute angle $BAC$ . Give your answer in degrees or radians.

(d) Find the distance from $A$ to the line $BC$ .

The coordinates of the points $P, Q, R$ and $S$ are $(4,1,-1)$ , $(3,3,5)$ , $(1,0,2c)$ and $(1,1,2)$ respectively. The vectors $\overrightarrow{QR}$ and $\overrightarrow{PR}$ are orthogonal.

(a) Find the value of $c$ .

(c) Hence, find the vector equation of the line $l$ which passes through the point $Q$ and is parallel to the vector $\overrightarrow{PR}$ .

The diagram below shows the parallelepiped $ABCDEFGH$ . The position vectors of $A, C, D$ and $E$ are

\begin{aligned} \mathbf{a} &= \mathbf{i} + 4\mathbf{j} - \mathbf{k} \\ \mathbf{c} &= \mathbf{i} + \mathbf{j} + 7\mathbf{k} \\ \mathbf{d} &= 2\mathbf{i} - \mathbf{j} + \mathbf{k} \\ \mathbf{e} &= -4\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \end{aligned}

in that order. The point $X$ lies on the line segment $[EC]$ and satisfies $\overrightarrow{EX}=\lambda \overrightarrow{EC}$ .

Find the least possible value of $|HX|$ , giving your answer exactly and to $3$ significant figures.

Let $\vec{v} = 3\vec{i}-\vec{j}+2\vec{k}$ and $\vec{c} = \vec{i}+5\vec{j}+\vec{k}$ .

(a) Show that $\vec{c}$ is perpendicular to $\vec{v}$ .

(b) Show that there is no vector $\vec{w}$ with $\left|\vec{w}\right| = 1$ for which $\vec{v}\times\vec{w} = \vec{c}$ .

(c) It is now given that $\left|\vec{w}\right| = 2$ and $\vec{v}\times\vec{w} = \vec{c}$ . Find the two possible values of the angle between $\vec{v}$ and $\vec{w}$ , each correct to three significant figures.

Line $L_1$ : $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}$ and plane $\pi$ : $2x - y + z = 5$ .

(a) Find the point of intersection of the line $L_1$ and the plane $\pi$ .

(b) Hence, $\overrightarrow{PS} \times \overrightarrow{PR}$ .

(b) if the line $L_1$ is parallel, perpendicular, or neither to the normal of the plane $\pi$ .

IB Mathematics Analysis and Approaches AHL 3.16—vector Product Questionbank

IB Mathematics Analysis and Approaches AHL 3.16—vector Product Questionbank

IB Mathematics Analysis and Approaches AHL 3.16—vector Product Questionbank

IB Mathematics Analysis and Approaches AHL 3.16—vector Product Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 3.16—vector Product Questionbank