Calc 3 Quiz Questions

What is the equation of the plane parallel to 3x+4y-z = 5 passing through the origin?

3(x-0) + 4(y-0) - 1(z-0) = 0

Find an equation of the plane P determined by the points (1,0,-1), (2,2,1), and (4,1,2)

4(x-1) + 3(y-0) - 5(z+1) = 0

Find the point at which the line parameterized by r(t) = <3, -4-2t, -4t> intersects the plane 3x - y + z = 7

<3, -10, -12>

Suppose that v has components <3,1>. How do the components change if you translate v horizontally 2 units to the left?

Components do not change with translation

Let a = <7,-7,1> and b = <2,3,-5>. Find a vector perpendicular to both a and b.

Cross product, use matrix method

Find a unit vector that is parallel in the opposite direction to v = <6,0,-8>

Find unit vector, invert sign to make opposite in direction

Suppose that a place with a normal vector n and a line with direction vector v both pass through the origin and that n 'dot' v = 0. Which of the following statements is correct? i. The line is contained in the plane ii. The line is orthogonal to the plane

The line is contained in the plane

True or False: if v is a direction vector for a line L, then -v is also a direction vector for L.

True

For each expression, determine if it is a scalar, vector, or undefined. (a) (u 'dot' v) x w (b) ( u x v) 'dot' w (c) ||w|| (u 'dot' v) (d) ||w|| (u x v)

Undefined Scalar Scalar Vector

What are the possible angles between two unit vectors e and f if ||e|| x ||f|| = 1/2

Unit vectors have magnitude of 1, so cross product equation is reduced to sin(theta) = 1/2. Hence, theta = pi/6 and 5pi/6

Let a = <7,-7,1> and b = <2,3,-5>. Is the angle between the two vectors acute, obtuse, right, all, or none?

Use dot product, answer is negative, hence obtuse.

Let a = <7,-7,1> and b = <2,3,-5>. Calculate the projection of a in the direction of b.

Use projection equation (parallel)

Provide clear sketches of the designated traces for the surface z^2 = (x/2)^2 + (y/4)^2 a). The trace in the plane z = 1 b). The trace in the plane x = 0

a). Oval b). 'X' flattened along the x axis

let u||v be the projection of u along v. Which of the following is the projection of u along the vector 2v? a). 1/2 u||v b). u||v c). 2u||v

b). u||v

The perpendicular component of v along v is a). v b). ev c). 0 d). undefined

c). 0

Which of the following statements are true of a plane that is parallel to the yz plane? i. n = <0,0,1> is a normal vector ii. n = <1, 0, 0> is a normal vector

n = <1,0,0> is a normal vector

Find a parameterization for the line that passes through P = (4,0,8) with direction vector v = 7i + 4j

r(t) = <4,0,8> + t<7,4,0>

Give an equation for a sphere of radius 3 centered at (0,0,-3)

x^2 + y^2 + (z-3)^2 = 9