Exam 12: Vectors and the Geometry of Space

a. Find an equation of the sphere that passes through the point $( 6 , - 6,3 )$ and has center $( - 4,6,4 )$ . b. Find the curve in which this sphere intersects the $x v$ -plane.

Calculate the angle between $\mathbf { a }$ and $\mathbf { b }$ (correct to the nearest degree). $\mathbf { a } = 3 \mathbf { i } - 2 \mathbf { j } , \mathbf { b } = - 7 \mathbf { j } + \mathbf { k }$

Two forces $F _ { 1 }$ and $F _ { 2 }$ with magnitudes $8 \mathrm { lb }$ and $12 \mathrm { lb }$ act on an object at a point $P$ as shown in the figure. Find the magnitude of the resultant force $F$ acting at $P$ . Round the result to the nearest tenth.

Find a nonzero vector orthogonal to the plane through the points $P , Q$ , and $R$ . $P ( 1,0,0 ) , Q ( 7,8,0 ) , R ( 0,8,1 )$

Find the distance between the point $( 9,9,3 )$ and the plane $15 x - 3 y - 8 z = 10$ . Round your answer to two decimal place.

A bicycle pedal is pushed by a foot with a force of $a - \mathrm { N }$ as shown. The shaft of the pedal is $12 \mathrm {~cm}$ long. Find the magnitude of the torque about $P$ correct to two decimal places. Let $a = 70 , b = 70 ^ { \circ } , c = 10 ^ { \circ }$ .

Let $\mathbf { v } = 7 \mathbf { j }$ and let $\mathbf { u }$ be a vector with length 5 that starts at the origin and rotates in the $x y$ - plane. Find the maximum value of the length of the vector $| \mathbf { u } \times \mathbf { v } |$ .

Find the midpoint of the line segment joining the given points. $( 4 , - 2 , - 3 )$ and $( 6 , - 4 , - 1 )$

Find the center and radius of the sphere. $x ^ { 2 } - 6 x - 26 + y ^ { 2 } + 24 y + z ^ { 2 } - 18 z = 0$

Identify the planes that are perpendicular. Select the correct answer.

An ellipsoid is created by rotating the ellipse $6 x ^ { 2 } + y ^ { 2 } = 36$ about the $x$ -axis. Find the equation of the ellipsoid.

Find the distance between the planes. $5 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 0$

Draw a rectangular box with the origin and $( 3,6,3 )$ as opposite vertices and with its faces parallel to the coordinate planes. Find the length of the diagonal of the box.

Which of the given lines is parallel to the line $x = 8 + t , y = t , z = - 7 - 10 t$ ?

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 2 x + 6 y = 0$

Find the length of each side of the triangle $A B C$ and determine whether the triangle is an isosceles triangle, a right triangle, both, or neither. Select the correct answer. $A ( - 1,0,1 ) , B ( 1,1 , - 1 ) , C ( 1,1,1 )$

Sketch the plane in a three-dimensional space represented by the equation. $z = - 1$