Exam 13: A Fundamental Tool- Vectors

Suppose that A, B, C, and D are four distinct points in 3-space.Then ( $\vec{AB}$ + $\vec{BC}$ )+ $\vec{CD}$ = $\vec{AD}$ .

Find a unit vector that points in the same direction as the displacement vector from the point P(0, 2, -3 )to the point Q(3, 5, 5).

If $\vec { w } = 2 \vec { i } + 5 \vec { j } + \alpha \vec { k }$ find a value of a such that $\| \vec { w } \| = 30$ .

For any vectors $\vec { u }$ and $\vec { v }$ , $\| \vec { u } + \vec { v } \| > \| \vec { u } - \vec { v } \|$ .

Consider the points P = (1, 2, 6), Q = (3, 4, 4)and R = (-1, 2, 12). Find a vector perpendicular to $\overrightarrow { P Q }$ and $\vec{PR}$ and then use it to find the equation of the plane which contains P, Q, and R.

If $\vec{u}, \vec{v}$ are non-zero vectors with $|\vec{z} \cdot \vec{v}|=\mid \overrightarrow{k z}\|\| \vec{v} \|$ , then $\vec { u }$ is either parallel or anti-parallel to $\vec { v }$ .

Consider the plane $5 x - y + 4 z = 20$ .Find the coordinates of the points where the plane intersects the three axes.

Suppose $\vec { \alpha }$ is a fixed vector of length 5, $\vec { b }$ is a vector which can rotate and has length 2, and $\theta$ is the angle between them. What is the maximum value of $\vec { a } \cdot \vec { b }$ , and for what value of $\theta$ does it occur?

For what value(s)of a is the vector $\vec { v } = - 50 \vec { i } + a \vec { j } + 5 \vec { k }$ perpendicular to the plane z = 10x - 2y + 7?

Find a unit vector in the same direction as ( $\vec{AB}$ + $\vec{BC}$ )+ $\vec{CD}$ = $\vec{AD}$ .Round your answer to 3 decimal places.

Suppose $\vec { u } = \vec { i } + 3 \vec { j } + 5 \vec { k }$ .Find the vector $\vec { v }$ of shortest length with $\vec { u } \cdot \vec { v } = 41$ .

A birdwatcher on the ground notices that a blue bird is heading north with a groundspeed of 28 miles/hr.The wind is blowing 25 miles/hr to the northeast.What is the airspeed of the bird?

Vectors $\vec { a } , \vec { b } , \vec { c }$ and $\vec { u }$ are shown in the picture below.

Shortly after takeoff, a plane is climbing with velocity vector $140 \vec { i } + 130 \vec { j } + 10 \vec { k }$ 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. If there is a wind blowing 20 km/hr to the southeast, what is the speed of the plane relative to the ground?

Consider the plane $3 x - 5 y + z = 5$ .Find the value of a that makes the vector $12 \vec { i } - 20 \vec { j } + \alpha \vec { k }$ perpendicular to the plane.

The unit sales of a company's product A in June, July, August, and September are given by $\vec { a } = ( 100,220,100,85 )$ , and the sales of product B in the same four months are given by $\vec { b } = ( 130,200,320,200 )$ .Suppose the company makes a profit of $200 for each unit of product A and $85 for each unit of product B.Compute $200 \vec { a } + 85 \vec { b }$ and interpret its meaning.

Three men are trying to hold a ferocious lion still for the veterinarian.The lion, in the center, is wearing a collar with three ropes attached to it and each man has hold of a rope.Charlie is pulling in the direction N 60° W with a force of 380 pounds and Sam is pulling in the direction N 44° E with a force of 400 pounds. What is the direction and magnitude of the force needed on the third rope to counterbalance Sam and Charlie?

An airplane wants to travel east, but there is a wind of 100 miles per hour blowing from the northwest.If the airspeed of the plane is 390 miles per hour, find the direction the plane should head (as an angle north of west)and its groundspeed.

If $\vec { u } = 2 \vec { i } + 2 \vec { j }$ and $\vec { v } = 2 \vec { i } - 2 \vec { j }$ , find the values $c _ { 1 }$ and $C _ { 2 }$ such that $18 \vec { i } + 6 \vec { j } = c _ { 1 } \vec { u } + c _ { 2 } \vec { v }$ .

If $\vec { v } = b \vec { w } \neq \overrightarrow { 0 }$ for some constant b, then the vectors $\vec { v }$ and $\vec { w }$ are parallel or anti-parallel.