Mathematics Extension 2

Consider three-dimensional vectors $\mathbf a=\overrightarrow{OA}$, $\mathbf b=\overrightarrow{OB}$ and $\mathbf c=\overrightarrow{OC}$ satisfying $\mathbf a\cdot\mathbf b=1$, $\mathbf b\cdot\mathbf c=2$, and $\mathbf c\cdot\mathbf a=3$. Which statement is true?

A. Two of $\mathbf a,\mathbf b,\mathbf c$ could be unit vectors.
B. The points $A,B,C$ could lie on a sphere centred at $O$.
C. For any three-dimensional vector $\mathbf a$, vectors $$ can be found satisfying the conditions. D. For all such vectors, there exist positive reals such that .

Using vector methods, show that the length of the perpendicular from $M$ to $\overrightarrow{ON}$ is

Find the angle between the vectors $\mathbf a=\mathbf i+2\mathbf j-3\mathbf k$ and $\mathbf b=-\mathbf i+4\mathbf j+2\mathbf k$, giving your answer to the nearest degree.

Find the equations of a sphere whose centre is at $(1,0,1)$ and touches the line

Using the diagram and information from part (e), calculate the angle between $\overrightarrow{MD}$ and $\overrightarrow{ON}$.

Another line, $\ell_2$, is defined by the vector equation

Find the possible values of $a$ when the angle between $\ell_1$ and $\ell_2$ is $\frac{\pi}{4}$.

What is the unit vector in the same direction as $\vec a=2\mathbf i+\mathbf j-3\mathbf k$?

A. $-\frac1{14}\begin{bmatrix}2\\1\\-3\end{bmatrix}$ B. C. D.

Let $A$ and $B$ be two distinct points in three-dimensional space. Let $M$ be the midpoint of $AB$.

Let $S_1$ be the set of all points $P$ such that $\overrightarrow{AP}\cdot\overrightarrow{BP}=0$.

Let $S_2$ be the set of all points $N$ such that $|\overrightarrow{AN}|=|\overrightarrow{MN}|$.

The intersection of $S_1$ and $S_2$ is the circle $S$.

What is the radius of the circle $S$?

A. $\dfrac{|\overrightarrow{AB}|}{2}$
B. ${\displaystyle }$ C. D.

Using the points defined in part (d), find $|\overrightarrow{CB}|$.

A triangle is formed in three-dimensional space with vertices $A(1,-1,2)$, $B(0,2,-1)$ and $C(2,1,1)$. Find the size of $\angle ABC$, giving your answer to the nearest degree.

Using the points defined in part (d), what is the size of the acute angle between $\overrightarrow{AB}$ and $\overrightarrow{CB}$? Answer correct to the nearest degree.

The two non-parallel vectors $\vec u$ and $\vec v$ satisfy $\lambda\vec u+\mu\vec v=\vec0$ for some real numbers $\lambda$ and $\mu$. Show that $\lambda=\mu=0$.

It is given that
$\vec c=\begin{bmatrix}1\\4\\0\end{bmatrix}+t\begin{bmatrix}-1\\1\\5\end{bmatrix}$
does not intersect $\vec r$. Point $T$ is on $\vec c$ and $S$ is a point on $\vec r$ such that $TS$ is a minimum. By first expressing $\vec c$ in parametric form, find an expression for $\overrightarrow{TS}$ in terms of $t$.

Show that the line
$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}15\\-1\\6\end{pmatrix}+\delta\begin{pmatrix}4\\-1\\-1\end{pmatrix}$
is also tangent to the sphere
$(x-5)^2+(y+2)^2+(z-3)^2=38$
by showing that it is perpendicular to the radius of the sphere at $\begin{pmatrix}7\\1\\8\end{pmatrix}$.

The line has vector equation
$\vec r=\lambda\begin{bmatrix}1\\3\\1\end{bmatrix}.$
For $P(x_1,y_1,z_1)$, the point $Q$ on the line is such that $PQ$ is a minimum. Using vector projection methods, show that
$\overrightarrow{OQ}=\frac{x_1+3y_1+z_1}{11}\begin{bmatrix}1\\3\\1\end{bmatrix}.$

Find the angle between the vectors $\vec a=\begin{pmatrix}2\\0\\4\end{pmatrix}$ and $\vec b=\begin{pmatrix}-3\\1\\2\end{pmatrix}$, giving the angle in degrees correct to 1 decimal place.

Given vectors $\vec a=\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}$ and $\vec b=\begin{pmatrix}b_1\\b_2\\b_3\end{pmatrix}$, show that
$|a_1b_1+a_2b_2+a_3b_3|\le\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}.$

The point $P(x,y,z)$ lies on the sphere of radius $1$ centred at the origin $O$. Using $\overrightarrow{OP}=x\vec i+y\vec j+z\vec k$ and the triangle inequality, or otherwise, show that
$|x|+|y|+|z|\ge1.$

As in part (i), $P(x,y,z)$ lies on the unit sphere centred at the origin. Using part (ii), or otherwise, show that
$|x|+|y|+|z|\le\sqrt3.$

Consider the two vectors $\mathbf u=-2\mathbf i-\mathbf j+3\mathbf k$ and $\mathbf v=p\mathbf i+\mathbf j+2\mathbf k$. For what values of $p$ are $\mathbf u-\mathbf v$ and $\mathbf u+\mathbf v$ perpendicular?