Dot Product

Orthogonal, Parallel, or Neither (TL 20)

100

Are〈2, 3〉and〈-6, 4〉orthogonal?

Yes, dot product is 0

200

Find k if a =〈k, 3〉and b =〈4, 8〉are orthogonal

No, dot product equals -6

300

What is the dot product of two parallel vectors (Pointing in the same direction) with magnitudes 3 and 7?

21

400

Determine the value of c such that the vectors a = 〈4, -6〉and b = 〈c, 9〉are parallel

c = -6

500

 If u and v are unit vectors and they are orthogonal, what is the magnitude of their sum, |u+v|?

sqrt(2)

Angles Between Vectors (TL 20)

100

Find the angle between the vectors a = ⟨1, 0⟩, b = ⟨0, 1⟩

90°

200

 If ab = -|a||b|, what is the angle between the vectors?

pi, 180°

300

Find the angle between the vectors  a = ⟨2, 1⟩, b = ⟨1, 2⟩

36.9°, cos-1(⅘), 0.644

400

Find the angle between the vectors a = i + 2j -2k, b = 4i - 3k

48.2°, cos-1(⅔), 0.841

500

A force vector acts as F = ⟨3, -2, 6⟩ and a displacement vector is d = ⟨1, 4, -2⟩. Find the angle between these two vectors.

122.0°, cos-1(-17721), 2.13

Projections (TL 45)

100

Given a = ⟨1, 0⟩ and b = ⟨7, 5⟩, what is the scalar projection of b onto a?

7

200

What is the scalar projection of b =〈3, 4〉onto a =〈5, 0〉?

3

300

Find the vector projection of b = ⟨4, 2⟩ onto a = ⟨1, 1⟩.

⟨3, 3⟩

400

Find the scalar projection of b = 〈-1, 4, 8〉onto a = 〈2, 1, 2〉

6

500

Find the vector projection of b = ⟨3, -2, 5⟩ onto a = ⟨2, 1, -2⟩. Simplify your answer completely.

⟨-4/3, -⅔, 4/3⟩

Direction Cosines and Magnitudes (TL 45)

100

Find the magnitude of the vector a = 3i + 5j

5

200

Find the magnitude of the vector a = -2i + 5j + 4k

3sqrt(5)

300

Find the direction cosines of the vector a = i + 2j + 2k

⟨1/3, 2/3, 2/3⟩

400

 Find the direction angles of the vector a = 2i - 2j + k

(alpha) = 48.2, (beta) =131.8, (gamma) =70.5

(alpha) = cos-1(⅔), (beta) =  cos-1(-⅔), (gamma) =  cos-1(⅓)

500

Find the direction cosines and direction angles of the vector a = -3i + 6j + 2k

⟨-3/7, 6/7, 2/7⟩  (alpha) = 115.4, (beta) = 31.0, (gamma) = 73.4

(alpha) = cos-1(-3/7), (beta) = cos-1(6/7), (gamma) = cos-1(2/7)

Dot Product Tricks and Applications (TL 30)

100

If the dot product is positive, the angle between vectors must be…

acute

200

If |u| = 4 and |v| = 7, what is the maximum possible value of u • v? (hint: what is the max value of cosine)

28

300

If you double the length of vector a, what will happen to dot product a • b?

It will double

400

A force vector〈5, 0〉moves an object along a displacement vector 〈10, 0〉. What is the work done?

50

500

A force F = 〈4, -3〉 acts on an object moving along displacement d = 〈6, 8〉.

Compute the work done and determine whether the force is helping, opposing, or perpendicular to the motion.

0; the work is perpendicular to motion