Exam 13: A Fundamental Tool- Vectors

Let $\vec { v } = 3 \vec { i } + 2 \vec { j } - 3 \vec { k }$ .Which of the following are parallel to $\vec { v } ?$ Select all that apply.

Let A(1, -2, -1), B(-1, -4, 0), C(0, -2, 1), and D(2, 0, 0)be the vertices of a parallelogram in that order. Find the area of the parallelogram to 4 decimal places.

Let $\vec { u }$ and $\vec { v }$ be two non-zero parallel vectors.Then $\vec { U } \cdot \vec { V } = \pm 1$ .

Let $\vec { v } = 2 \vec { i } - 2 \vec { j } + 3 \vec { k }$ and let $\vec { W }$ be a vector of magnitude 6 at an angle of 30° to $\vec { v }$ .Find $\vec { v } \cdot \vec { w }$ .

Shortly after takeoff, a plane is climbing with velocity vector $150 \vec { i } + 130 \vec { j } + 12 \vec { k }$ Assume the units are kilometer per hour, that the x-axis points east, that the y-axis points north, that the z axis points up.What is the airspeed of the plane?

Find the missing numbers. $( 4 \vec { i } + \ldots \vec { j } + 8 \vec { k } ) \times ( \vec { i } - \vec { j } + \vec { k } ) = \underline { i } + 4 \vec { j } - 6 \vec { k }$

Find the equation of the plane parallel to $4 \vec { i } - \vec { j } + 3 \vec { k }$ and $3 \vec { i } + 3 \vec { j } - \vec { k }$ and containing the point (1, 1, 2).

Using vectors, show that if $\| \vec { u } \| = \| \vec { v } \|$ , then $\vec { U } + \vec { v }$ and $\vec { U } - \vec { v }$ are perpendicular.Conclude that the diagonals of a rhombus are perpendicular.

Find the missing number. $\left( 5 \vec { i } + \vec { j } ^ { - } + 4 \bar { k } \right) \cdot ( \vec { i } - \vec { j } + \bar { k } ) = 15$

Consider two vectors $\vec { v } = - 3 \vec { i } - \vec { j } + 3 \vec { k }$ and $\vec { w } = a \vec { i } + a \vec { j } - \vec { k }$ .For what values of a are $\vec { v }$ and $\vec { W }$ parallel?

The vectors $\vec { v }$ and $\vec { W }$ are shown below.Determine if the following statement is true or false. $( \vec { v } - \vec { w } ) \cdot \vec { j } > 0$

Let $\vec { v } = 6 \vec { i } + 3 \vec { j } - 6 \vec { k }$ .Considering $- \vec { v }$ as a position vector that begins at the origin, at what point of three space does its head lie?

Let $\vec { u } = - 4 \vec { i } + 2 \vec { j } + 3 \vec { k }$ and $\vec { v } = \vec { i } - 2 \vec { j } + 6 \vec { k }$ .Compute $\vec { u } \cdot \vec { v }$

The plane $x + 2 y + z = 0$ has a normal $\vec { n } = \vec { i } + 2 \vec { j } + \vec { k }$ Let $\vec { u } = \vec { i } + \vec { j } + \vec { k }$ .Express $\vec { u }$ as the sum of a vector $\vec { v }$ which is parallel to $\vec { n }$ and a vector $\vec { w }$ which is parallel to the plane.

The triangle ABC with vertices $A = \vec { i } + 5 \vec { j }$ , $B = \vec { i } - 3 \vec { j }$ , and $C = - 2 \vec { i } + 2 \vec { j }$ is a right triangle.

Given that $\vec { v }$ and $\vec { w }$ are non-zero vectors and that $\vec { v } \cdot \vec { w } = 0$ and $\vec { v } \times \vec { w } = \vec { i } + \vec { j } + \vec { k }$ , then which of the following MUST be true? Select all that apply.

Determine the distance from the plane $2 x + y - z = 1$ to the plane $2 x + y - z = 6$ .Give your answer to 4 decimal places. (Hint: Although there are many approaches, projections come in handy.)

For what value(s)of a is the vector $\vec { v } = 3 \vec { i } + a \vec { j } + 3 \vec { k }$ parallel to the plane $z = 2 x - 5 y + 2$ ?

Consider the plane $4 x - 3 y + z = 3$ .What is the value of a if the vector $3 \vec { i } - 5 \vec { j } + a \vec { k }$ is parallel to the plane?