Exam 13: A Fundamental Tool- Vectors

If $\vec { v } = 2 \vec { i } + \vec { j } + 4 \vec { k }$ and $\vec { w } = \vec { i } + \vec { j } - 2 \vec { k }$ , find the value of s so that $\vec { v } + s \vec { w }$ is perpendicular to $\bar { w }$ .

If $\vec { u } = 4 \vec { i } + 5 \vec { j } + 3 \vec { k }$ and $\vec { v } = 2 \vec { i } - 4 \vec { j } + 3 \vec { k }$ , find $5 \vec { u } + 4 \vec { v }$ .

Let ABCDEF be a regular hexagon.Express the vectors $\overrightarrow { D \vec { E } }$ and $\overrightarrow { C \vec { T } }$ in terms of $\vec{AB}$ .

If $\vec { a } \neq \overrightarrow { 0 }$ , then $\frac { \vec { a } } { \| \vec { a } \| }$ is a unit vector which is parallel to $\vec { a }$ .

If $\vec { w } = 2 \vec { i } - \vec { j } + 2 \vec { k } , \quad \vec { v } = \vec { i } + 3 \vec { j }$ find a unit vector perpendicular to both $\vec { W }$ and $\bar { v } .$

Suppose that $\vec { u }$ and $\vec { v }$ are non-zero vectors and $( \vec { u } + \vec { v } ) \times ( \vec { u } - \vec { v } ) = \overrightarrow { 0 }$ .Then what MUST be true about $\vec { u }$ and $\vec { v }$ ?

If $\|a \vec{i}+b \vec{j}\|>\|c \vec{i}+d \vec{j}\|$ , then a > b and c > d.

Find the vector $\vec { u }$ in 3-space which satisfies both of the following conditions. (a) $\vec { u } \cdot \vec { k } = 5$ (b) $\vec { u } \times \vec { k } = 5 \vec { i } - 2 \vec { j }$

Find the angle between the planes $- 4 ( x - 1 ) + 3 ( y + 2 ) + 5 z = 0$ and $x + 4 ( y - 1 ) + 2 ( z + 4 ) = 0$ .

If the vector $2 \vec { i } - 5 \vec { j } + \alpha \vec { k }$ has magnitude 9, find a.

If $\vec { a } , \vec { b } \neq \overrightarrow { 0 }$ , then $\vec { a } \cdot ( \vec { a } \times \vec { b } ) = 0$ .Why is this true?

Two boats leave a harbor at the same time.Boat A cruises northwest at a rate of 17 knots (nautical miles per hour).Boat B cruises north at a rate of 21 knots. (a)Find the displacement vector from boat A to boat B half an hour later. (b)For the passengers in boat A, what does the velocity of boat B appear to be?

Let ABCDEF be a regular hexagon.Express $\frac { 1 } { 2 } \overrightarrow { A D } - \overrightarrow { A B }$ in terms of $\vec {B E }$ .

Compute the area of the triangle with vertices A(0, 0, 0), B(3, 3, 0), and C(3, 6, 3).

What is the geometric definition of $\| \vec { v } \times \vec { w } \|$ ?

Let ABCDEF be a regular hexagon.Express $\frac { 1 } { 2 } \overrightarrow { A D } + \overrightarrow { AE }$ in terms of $\vec {D F }$ .

Let A, B, C, D, E, and F be points on a plane.Explain why .

Given that $\vec { v }$ and $\vec { w }$ are non-zero vectors and that $\vec { v } \cdot \vec { w } = 2$ , then which of the following MUST be true? Select all that apply.

Find the volume of the parallelopiped with edges $4 \vec { i } - 3 \vec { j } + \vec { k }$ , $\vec { i } - 6 \vec { j } + \vec { k }$ , and $\vec { i } + \vec { j } + 2 \vec { k }$ .

The vector $\vec { a } \times \vec { b }$ is parallel to the x-axis where $\vec{a}$ and $\vec { b }$ are shown below (both $\vec { \alpha }$ and $\vec { b }$ are in the xy-plane).