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MATH115 GEOMETRY AND CALCULUS

Quiz 1

1.

Normal of a plane Let $W$ be the plane in 3-space that goes through the points $P=(2,1,-2)$ , $Q=(1,1,3)$ and $R=(-1,0,2)$ . A normal of $W$ is given by

A) $(0\;\;-4\;\;4)$ ; $\quad$ B) $(2\;\;8\;\;2)$ ; $\quad$ C) $(-2\;\;-1\;\;0)$ ; $\quad$ D) $(5\;\;-11\;\;1)$ ; $\quad$ E) $(3\;\;3\;\;3)$ .
2.

Coordinates If $A B C D$ is a parallelogram, then the coordinates of the point $D$ are

A) $(2\;\;-1\;\;0)$ ; $\quad$ B) $(4\;\;2\;\;3)$ ; $\quad$ C) $(3\;\;5\;\;3)$ ; $\quad$ D) $(5\;\;4\;\;2)$ ; $\quad$ E) $(2\;\;-3\;\;1)$ .
3.

Area The area of the parallelogram $A B C D$ is

A) $\sqrt{27}$ ; $\quad$ B) 5; $\quad$ C) $\sqrt{23}$ ; $\quad$ D) $\sqrt{17}$ ; $\quad$ E) $\sqrt{26}$ .
4.

Vector operations Which of the following statements about vectors ${\bf u}$ , ${\bf v}$ in ${\mathbb{R}}^{n}$ is false?

A) ${\bf u}\cdot({\bf v}+{\bf w})={\bf u}\cdot{\bf v}+{\bf u}\cdot{\bf w}$ ; $\quad$ B) ${\bf u}\times({\bf v}\times{\bf w})=({\bf u}\times{\bf v})\times{\bf w}$ ; $\quad$ C) ${\bf u}\times({\bf v}+{\bf w})={\bf u}\times{\bf v}+{\bf u}\times{\bf w}$ ; $\quad$ D) $(-{\bf u})\times{\bf v}={\bf u}\times(-{\bf v})$ ; $\quad$ E) $({\bf u}\times{\bf v})\cdot{\bf u}=0$ . $\quad$
5.

Vector analysis Which of the following statements about vectors ${\bf u}$ , ${\bf v}$ in ${\mathbb{R}}^{n}$ is false?

A) ${\bf u}\cdot({\bf u}+2{\bf v})=|{\bf u}+{\bf v}|^{2}-|{\bf v}|^{2}$ ; $\;\;\;$ B) If ${\bf u}\cdot{\bf v}>0$ then $|{\bf u}|\,|{\bf v}|>0$ ;

C) $({\bf u}-{\bf v})\cdot({\bf u}+{\bf v})\leq|{\bf u}|^{2}$ ; $\;\;\;$ D) $|{\bf u}+{\bf v}|^{2}+|{\bf u}-{\bf v}|^{2}=2|{\bf u}|^{2}+2|{\bf v}|^{2}$ ;

E) If ${\bf u}$ and ${\bf v}$ are non-zero vectors then $|{\bf u}+{\bf v}|^{2}>0$ .

Assessed Exercises 1

1.

(2 marks) Let $\mathbf{u}$ and $\mathbf{v}$ be two vectors in $\mathbb{R}^{n}$ . Show that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal vectors if and only if $|\mathbf{u}+\mathbf{v}|^{2}=|\mathbf{u}-\mathbf{v}|^{2}$ .
2.

(3 marks) Let $\mathbf{u}$ and $\mathbf{v}$ be two vectors in $\mathbb{R}^{n}$ . Use the Cauchy-Schwarz inequality to prove the triangle inequality for vectors: $|{\bf u}+{\bf v}|\leq|{\bf u}|+|{\bf v}|$ .
3.

(2 marks) Find the angle between a diagonal of a cube and one of its edges.
4.

(3 marks) (i) Show that if ${\bf u}$ , ${\bf v}$ are vectors in ${\mathbb{R}}^{n}$ and ${\bf v}$ is non-zero, then ${\bf u}-\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}}{\bf v}$ is orthogonal to ${\bf v}$ .

(ii) Draw a diagram of ${\bf u}$ , ${\bf v}$ and ${\bf u}-\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}}{\bf v}$ .