One number that knows the angle

Mathematics · Geometry · ages 19-20

Name ______________________ Date ____________

What is the angle between (1, 1) and (0, 3)?
- 30 degrees
- 45 degrees
- 60 degrees
- 90 degrees
Compute the dot product (3, -2, 1) dot (2, 4, 5).
Answer: ______________
Compute the dot product (1, 2) dot (3, 4).
Answer: ______________
Compute the dot product (2, 3) dot (-1, 4).
Answer: ______________
For which value of k is (2, k) perpendicular to (3, 6)?
Answer: ______________
The dot product of two nonzero vectors is -6. What can you say about the angle between them?
- It is under 90 degrees
- It is exactly 90 degrees
- It is over 90 degrees
Let u = (2, -1, 3) and v = (1, 5, 1). Priya says u and v are perpendicular. Is she right?
Circle one: True False
What is the scalar component of (1, 2) along the direction (3, 4)?
Answer: ______________
What is the projection vector of (4, 0) onto the direction (1, 1)?
- (2, 2)
- (4, 4)
- (2, 0)
Mia says two nonzero vectors with dot product zero must be perpendicular. Is Mia right?
Circle one: True False

Answer key

For grown-ups. Fold this page away before handing over the rest.

45 degrees · The dot is 3 and the lengths are root 2 and 3, so the cosine is 1 over root 2, which gives 45 degrees.
3 · Multiply matching components and add: 6 - 8 + 5 = 3.
11 · Multiply matching parts and add: 3 + 8 = 11.
10 · 2 times -1 plus 3 times 4 is -2 + 12 = 10.
-1 · The dot is 6 + 6k, which is 0 when k = -1.
It is over 90 degrees · A negative dot means a negative cosine, and cosine is negative only for angles over 90 degrees.
True · The dot is 2 - 5 + 3 = 0, and both vectors are nonzero, so they are perpendicular.
2.2 · Normalize (3, 4) to (3/5, 4/5), then dot: 3/5 + 8/5 = 11/5 = 2.2.
(2, 2) · Dot with (1, 1) to get 4, divide by squared length 2, and scale (1, 1) by 2.
True · Zero dot with nonzero lengths gives cosine 0, which is exactly 90 degrees.