Points and Vectors

9000108704

Level:

Consider a pair of vectors $\vec{u} = (1,0,-1)$ and $\vec{v} = (2,-1,1)$. Find all the vectors $\vec{w}$ which are perpendicular to both $\vec{u}$ and $\vec{v}$ and satisfy $\left |\vec{w}\right | = 2$.

$\vec{w} = \left (\frac{2\sqrt{11}} {11} , \frac{6\sqrt{11}} {11} , \frac{2\sqrt{11}} {11} \right )$, $\vec{w} = \left (-\frac{2\sqrt{11}} {11} ,-\frac{6\sqrt{11}} {11} ,-\frac{2\sqrt{11}} {11} \right )$

$\vec{w} = (-1,-3,-1)$, $\vec{w} = (1,3,1)$

$\vec{w} = \left (-\frac{1} {2},-\frac{3} {2},-\frac{1} {2}\right )$, $\vec{w} = \left (\frac{1} {2}, \frac{3} {2}, \frac{1} {2}\right )$

$\vec{w} = \left (\frac{2\sqrt{2}} {3} , \frac{3\sqrt{2}} {2} , \frac{2\sqrt{2}} {3} \right )$, $\vec{w} = \left (-\frac{2\sqrt{2}} {3} ,-\frac{3\sqrt{2}} {2} ,-\frac{2\sqrt{2}} {3} \right )$

9000108705

Level:

Given vectors $\vec{u} = (1,y,3)$ and $\vec{v} = (1,2,1)$, find the value of the parameter $y$ which ensures that the vectors are perpendicular.

$- 2$

$1$

$2$

$0$

9000108706

Level:

Find all vectors which are parallel to the vector $\vec{u} = (3,-1)$ and have the length equal to $1$.

$\left (\frac{3\sqrt{10}} {10} ,-\frac{\sqrt{10}} {10} \right )$, $\left (-\frac{3\sqrt{10}} {10} , \frac{\sqrt{10}} {10} \right )$

$(0,-1)$, $(0,1)$

$(-3,1)$, $(3,-1)$

$\left (\frac{3} {4},-\frac{1} {4}\right )$, $\left (-\frac{3} {4}, \frac{1} {4}\right )$

9000108801

Level:

Find the angle between the vector $\vec{u} = (1,\sqrt{2})$ and $\vec{v} = (3,-1)$. Round to the nearest degree.

$73^{\circ }$

$42^{\circ }$

$57^{\circ }$

$64^{\circ }$

9000108802

Level:

Given the points $A = [1,2]$, $B = [2,6]$ and $C = [3,-1]$, find the interior angles of the triangle $ABC$. Round to the nearest degree.

$22^{\circ }$, $26^{\circ }$, $132^{\circ }$

$26^{\circ }$, $45^{\circ }$, $109^{\circ }$

$22^{\circ }$, $48^{\circ }$, $110^{\circ }$

$17^{\circ }$, $31^{\circ }$, $132^{\circ }$

9000108803

Level:

Consider the vector $\vec{u} = (\sqrt{3},1)$. Find the vector $\vec{w}$ such that $\left |\vec{w}\right | = 4$ and the angle between $\vec{u}$ and $\vec{w}$ is $60^{\circ }$. Find all solutions.

$\vec{w} = (0,4)$, $\vec{w} = (2\sqrt{3},-2)$

$\vec{w} = (0,-4)$, $\vec{w} = (\sqrt{7},-3)$

$\vec{w} = (0,4)$, $\vec{w} = (\sqrt{7},3)$

$\vec{w} = (\sqrt{5},\sqrt{11})$, $\vec{w} = (2\sqrt{3},-2)$

9000108804

Level:

The point $A = [3,2]$ is rotated about the center $B = [1,1]$ by $60^{\circ }$. Find the coordinate of its final position. Consider both clockwise and counterclockwise direction.

$\left [2\pm \frac{\sqrt{3}} {2} , \frac{3} {2} \mp \sqrt{3}\right ]$

$\left [1\pm \frac{\sqrt{3}} {2} , \frac{1} {2} \mp \sqrt{3}\right ]$

$\left [2\pm \frac{\sqrt{2}} {2} , \frac{3} {2} \mp \sqrt{2}\right ]$

$\left [1\pm \frac{\sqrt{2}} {2} , \frac{1} {2} \mp \sqrt{2}\right ]$

9000108805

Level:

Find the angle between $\vec{u} = (1,-2,3)$ and $\vec{v} = (-1,0,2)$. Round to the nearest degree.

$53^{\circ }$

$27^{\circ }$

$60^{\circ }$

$46^{\circ }$

9000108806

Level:

Find the coordinate $y$ which ensures that the vectors $\vec{u} = (-6,y,3)$ and $\vec{v} = (12,4,4)$ are perpendicular.

$15$

$12$

$\sqrt{5}$

$\frac{5} {3}$

9000108807

Level:

Find the angle between the median $t_{c}$ and side $c$ in the triangle $ABC$ for $A = [1,2]$, $B = [7,-2]$ and $C = [6,1]$. Round to the nearest degree. Hint: In geometry, the median $t_{c}$ of the triangle $ABC$ is the line segment joining the vertex $C$ to the midpoint of the opposing side.

$60^{\circ }$

$50^{\circ }$

$43^{\circ }$

$71^{\circ }$

9000108704

9000108705

9000108706

9000108801

9000108802

9000108803

9000108804

9000108805

9000108806

9000108807

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