All about Vectors

1

100

This is the component form of the sum of vectors i and j.

What is the vector <1,1>?

200

A directed line segment in the plane.

What is a vector in 2 dimensions?

300

This is the sum of vectors <1,1> and <-1,1> in linear combination form.

What is 2j?

400

True or False: If u is a unit vector, then 1/u is also a unit vector.

False: 1/u is not a vector.

500

Find the vector with tail at (-2,2,3) and tip at (4,0,-3) in terms of i, j, and k.

-6i+2j+6k

2

100

How do you make a vector a unit vector?

Divide each component of the vector by its length

200

Fill-in-the-blank: A vector whose length is 1 is called a(n) _________ vector.

Unit vector

300

This is vector multiplication of two vectors that produces a scalar.

What is the dot product?

400

This is vector multiplication of two vectors that produces another vector.

What is the cross-product?

500

True or False: 

If u is any vector, then u = (u dot i)*i+(u dot j)*j

True! Can you explain why?

3

100

What is the angle between two vectors that are parallel?

0 or 180 degrees

200

Describe the dot product between the following two vectors: 

u = <a, b> and v = <c, d>

u dot v = a*c+b*d

300

True or False: If u is a unit vector, then u dot u = 1. Justify your answer!

True! u dot u = |u|^2 = (1)^2 = 1

400

Fill-in-the-blank: If the angle between two vectors is pi/2, then their dot product is _____ .  

zero

500

Find the equation of a sphere with center at (0,4,-2) and radius 5.

x^2 + (y-4)^2 + (z+2)^2=25

4

100

True or False: If u is a unit vector, then so is -u.

True! Multiplying by -1 does not change the length but it does flip direction!

200

For any two vectors u and v in what dimension, there exists another vector that is orthogonal to both u and v.

What is the third dimension?

300

True or False: u dot 0 = 0

False! The dot product yields a scalar! 

so: u dot 0 = 0

400

Describe how to find the cross product of u =<a, b> and v =<c, d>.

TRICK QUESTION: we have only defined the cross product for 3D vectors!

500

Give the magnitude of the following vector:

u = 2i-3j

|u| = sqrt(13)

Misc.

100

True or False: vectors are defined by their magnitude only.

False: vectors are quantities that have direction and magnitude.

200

T or F: Force is a physical example of a vector.

True!

300

List three unit vectors that are orthogonal to each other.

EX: ihat, jhat, khat

400

Find the length of u + length of v if both u and v are unit vectors and the angle between them is 90 degrees.

2

500

Describe how to find the cross product of 

u =<a, b, c> and v =<d, e, f>.

23,31,12 -OR- <bf-ce,cd-af,ae-bd>