Exam 13: A Fundamental Tool- Vectors

There are exactly two unit vectors perpendicular to the vector $\vec { i } + 2 \vec { j } + \vec { k }$ .

Given the points P = (1, 2, 16), Q = (3, 5, 35), and R = (2, 5, 33), find the (perpendicular)distance from the point R to the line through P and Q.Give your answer to 4 decimal places.

The angle between $4 \vec { i } + 2 \vec { k }$ and $\vec { i } - 2 \vec { j } + 5 \vec { k }$ is less than $\pi$ /2.

An object, P, is pulled by a force of magnitude 16lb at an angle of 20 degrees north of east.In what direction must a force of magnitude 10 lb pull to ensure that P moves due east?

One of the highlights of the Calaveras Fair, besides the bullfrog jumping contest, is the watermelon seed spitting contest.The seed spitting is being done in the direction of $\vec { s } = 5 \vec { i } + \vec { j }$ .The wind velocity during the contest is $\vec { w } = \vec { i } + 4 \vec { j }$ mph.The rules say that a legal wind in the direction of the seed spit must not exceed 3 mph.Mr.Samuel C.himself just spit for a record of 22 ft 1.4 inches. Will his record stand?

Consider the plane $- 2 x + 2 y + 2 z = 0$ Find, to three decimal places, the shortest distance from the plane to the point $P = ( - 1,1,2 )$

Find a vector of length 5 that points in the same direction as the displacement vector from the point P(-1, 2, 2)to the point Q(3, 6, 5).

Consider the point $P = ( R \cos \theta , R \sin \theta )$ , which lies on the circle of radius R centered at the origin.Show that, so long as $\theta \neq 0$ or $\pi$ radians, then the vector from (R,0)to P and the vector from (-R,0)to P are perpendicular.

Determine a formula for the distance between a point (a, b, c)and the y-axis.

Consider two vectors $\vec { v } = 5 \vec { i } - \vec { j } + 3 \vec { k }$ and $\vec { w } = a \vec { i } + a \vec { j } - \vec { k }$ .For what value(s)of a are $\vec { v }$ and $\vec { W }$ perpendicular?

If $\vec { w } = 5 \vec { i } - \vec { j } + 5 \vec { k } , \quad \vec { u } = 2 \vec { i } + a \vec { j } + 5 \vec { k }$ find the value of a making $\vec { W }$ and $\vec { u }$ perpendicular.

Given that $\vec { v }$ and $\vec { w }$ are non-zero vectors and that $( \vec { V } \times \vec { w } ) \cdot \vec { u } = 0$ for all vectors $\vec { u }$ , then which of the following MUST be true? Select all that apply.

If the vector $2 \vec { i } + \alpha \vec { k }$ is parallel to $7 \vec { i } + b \vec { j } + 35 \vec { k }$ find a and b.

Suppose that $\vec { a }$ and $\vec { b }$ are non-zero vectors.If $\vec { a }$ and $\vec { b }$ are perpendicular, then which of the following MUST be true? Select all that apply.

Suppose that $\vec { a }$ and $\vec { b }$ are non-zero vectors.If $\vec { a }$ and $\vec { b }$ are parallel, then which of the following MUST be true? Select all that apply.

Suppose that A, B, C, and D are four distinct points in 3-space.Which of the following must be true? Mark all correct answers.

Suppose that $\vec { u }$ and $\vec { v }$ are non-zero vectors, and that $\vec { u }$ is perpendicular to $\vec { v }$ .Let $\vec { w } = \vec { u } \times ( \vec { v } \times \vec { u } )$ .Which of the following statements are true? Select all that apply.

A surveyor is standing at point A, 403 feet away from a tall building.Let $\vec { i } , \vec { j } , \vec { k }$ be the direction of east, north and up respectively.She observes that a vector in the direction of $\overrightarrow { A C }$ is $12 \vec { i } + 5 \vec { j } + 15 \vec { k }$ , where C is the highest point of the building. What is the height of the building?

A company manufactures four grades of videotape: Normal, Extra, Super, and Supreme.Last year, the company sold a₁ boxes of Normal tapes for $5.00 each, a₂ boxes of Extra tapes for $5.50 each, a₃ boxes of Super for $6.00 each, and a₄ boxes of Supreme for $8.00 each.Let $\vec { a } = \left( a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 } \right)$ , $\vec { b } = ( 5,5.5,6,8 )$ , and $\vec { C } = ( 2,2.4,2.5,3 )$ . (a)Compute $\vec { a } \cdot \vec { b } - \vec { a } \cdot \vec { C }$ . (b)If the computation in part (a)represents the company's profit last year, then what does the vector $\vec { C }$ represent?

A plane wants to go due Northeast, but a wind is blowing toward the North at 78 miles per hour.If the airspeed of the plane is 250 miles per hour, which direction should the plane head? What will be its groundspeed?