IB Mathematics Analysis and Approaches Topic AHL…

Let $\mathbf{a}=(2,-1,3),\qquad \mathbf{b}=(k,4,-2),\qquad \mathbf{c}=(1,2,1),$ where $k\in\mathbb{R}$ . Define the scalar triple product by $[\mathbf{a},\mathbf{b},\mathbf{c}]=\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$ .

Find the value of $k$ for which $\mathbf{a}$ and $\mathbf{b}$ are perpendicular.

For this value of $k$ , find the exact angle between and .

Find the angle between the following vectors $\mathbf{a}= =-4\mathbf{i}-2\mathbf{j}$ and $b$ , giving your answer in exact form.

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

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

Let $\vec{u}$ and $\vec{v}$ be vectors with and , and let be the angle between them.

For $k=5$ , show that the three vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ are not coplanar.

Find a unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ (for $k=5$ ). Give your answer in exact surd form.

Prove that the angle between $\mathbf{a}\times\mathbf{b}$ and $\mathbf{c}$ satisfies
$\cos\phi=\frac{[\mathbf{a},\mathbf{b},\mathbf{c}]}{|\mathbf{a}\times\mathbf{b}|\,|\mathbf{c}|}.$

Find $\vec{a}\cdot\vec{b}$ and $\left|\vec{b}\right|$ .

Find the value of $\lambda$ for which $\vec{a} - \lambda\vec{b}$ is perpendicular to .

For this value of $\lambda$ , find $\left|\vec{a} - \lambda\vec{b}\right|$ in exact form.

Find the exact value of $\cos\theta$ , where $\theta$ is the angle between $\vec{a}$ and .

Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .

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

Find the unit vector in the direction $\overrightarrow{AC}-\overrightarrow{AB}$ .

Find the distance from $A$ to the line $BC$ .

Find expressions for $\left|\vec{u}+\vec{v}\right|^{2}$ and $\left|\vec{u}-\vec{v}\right|^{2}$ in terms of $\theta$ .

Given that $\left|\vec{u}+\vec{v}\right| = 10$ , find the value of , correct to three significant figures.

For this value of $\theta$ , find the angle between $\vec{u}+\vec{v}$ and , correct to three significant figures.

Find the two values of $\theta$ for which the angle between $\vec{u}+\vec{v}$ and is radians, each correct to three significant figures.

IB Mathematics Analysis and Approaches AHL 3.13—scalar (dot) Product Questionbank

IB Mathematics Analysis and Approaches AHL 3.13—scalar (dot) Product Questionbank

IB Mathematics Analysis and Approaches AHL 3.13—scalar (dot) Product Questionbank

IB Mathematics Analysis and Approaches AHL 3.13—scalar (dot) Product Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 3.13—scalar (dot) Product Questionbank