Exam 13: A Fundamental Tool- Vectors

Given the points P = (1, 2, 10), Q = (3, 5, 24), and R = (2, 5, 20), find the equation of the plane containing P, Q, and R.

The vector $5 \vec { i } - 6 \vec { j } + 6 \vec { k }$ is the displacement vector from the point (1, -1, 3)to which other point?

For any vectors $\vec { u }$ and $\vec { v }$ , $( \vec { u } + \vec { V } ) \cdot ( \vec { u } - \vec { v } ) = \| \vec { u } \| ^ { 2 } - \| \vec { v } \| ^ { 2 }$ .

Consider the plane $- 3 x - 5 y + 4 z = 0$ Find a normal vector $\vec { n }$ to this plane.

If $\vec { a } , \vec { b } \neq \overrightarrow { 0 }$ , then find a vector which is perpendicular to $\vec { a }$ and $\vec { b }$ .

An airplane is descending in preparation to land at an airport to its northwest.Its airspeed is 640 km/hr and its altitude is decreasing a rate of 50 km/hr.Describe its velocity vector.

Let A(-2, 1, -1), B(-3, 2, 0), C(-2, 1, 1), and D(-1, 0, 0)be the vertices of a parallelogram in that order. Find the area of the projection of the parallelogram in the xy-plane.

Determine the equation of the plane which contains the points (1, 2, -8)and (-2, 1, -8)and is perpendicular to the plane $x + 3 y - 8 z = 2$ .

A bird is flying with velocity $\vec { v } = 8 \vec { i } + 3 \vec { j }$ (measured in m/ sec )relative to the air.The wind is blowing at a speed of 4 m/ sec parallel to the x-axis but opposing the bird's motion.Find the speed of the bird relative to the ground.

Find the missing numbers.The vector $\vec { i } + \vec { j } + \vec { k }$ has magnitude 3 and is pointing in the direction of the vector $\vec { i } - 8 \vec { j } - 8 \vec { k }$ .

Let $\vec { v } = 4 \vec { i } - 4 \vec { j } + 3 \vec { k }$ and let $\vec { W }$ be a vector of magnitude 4 at an angle of 30° to $\vec { v }$ .Find $\| \vec { v } \times \vec { w } \|$ .

Determine two vectors of length 7 in the xy-plane that are normal to the parabola $y = x ^ { 2 }$ at (2, 4).

Determine the distance from the plane $- 2 x + 1 y + 4 z = 1$ to the point $P = ( - 1 , - 3,4 )$ Give your answer to 4 decimal places.

The vector $\vec { a } \times \vec { b }$ has a positive z component where $\vec{a}$ and $\vec { b }$ are shown below (both $\vec { \alpha }$ and $\vec { b }$ are in the xy-plane).

Given the points P = (1, 2, 9), Q = (3, 5, 26), and R = (2, 5, 22), find the area of the triangle PQR.Give your answer to 4 decimal places.

Find a unit vector in the direction of the vector $0.4 \vec { i } - 0.8 \vec { j } + 1.5 \vec { k }$ .

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

Find the missing numbers: $3 ( a \vec { i } + b \vec { j } + 3 \vec { k } ) - 3 ( - \vec { i } + \vec { j } + c \vec { k } ) = \vec { i } + 4 \vec { j } - 5 \vec { k }$

Even without knowing the vector $\vec { v }$ , you can tell that the following statement must be wrong.Explain why. $( 4 \vec { i } - 9 \vec { j } + 8 \vec { k } ) \times \vec { v } = ( - 4 \vec { i } + 4 \vec { j } + 3 \vec { k } )$

Does the point D(-3, 4, -6)lie in the same plane as the triangle with vertices A(0, 0, 0), B(5, 5, 35), C(5, 4, 33)?