Lesson test: Vectors

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Vectors

1.[1p]
What is the length of the vector $(2, 3, 6)$ ?

CorrectNot quite: 7

2.[1p]
What is $(1, 2) \cdot (3, 1)$ ?

CorrectNot quite: 5

3.[2p]
Two non-zero vectors are perpendicular exactly when

their dot product is zero

Correct

their dot product is one

The answer is: their dot product is zero

their lengths are equal

The answer is: their dot product is zero

one is a scalar multiple of the other

The answer is: their dot product is zero

4.[2p]
What is the angle, in degrees, between $(1, 2)$ and $(3, 1)$ ?

CorrectNot quite: 45

5.[1p]
A negative dot product means the angle between the two vectors is obtuse.

True

Correct

False

The answer is: True

6.[2p]
Which of these are operations that the definition of a vector space provides?

Select all that apply

Adding two vectors componentwise

Correct

Multiplying a vector by a scalar

Correct

Dividing one vector by another

The answer is: Adding two vectors componentwise, Multiplying a vector by a scalar

Multiplying two vectors to give a third vector in the same space

The answer is: Adding two vectors componentwise, Multiplying a vector by a scalar

7.[3p]
Projecting $𝐛 = (4, 3)$ onto $𝐚 = (2, 1)$ gives $c 𝐚$ . What is $c$ ?

CorrectNot quite: 2.2

8.[2p]
The formula $𝐚 \cdot 𝐛 = ‖ 𝐚 ‖ ‖ 𝐛 ‖ \cos θ$ is obtained by

expanding $‖ 𝐛 - 𝐚 ‖^{2}$ two ways and comparing it with the law of cosines

Correct

defining the angle to make it true

The answer is: expanding $‖ 𝐛 - 𝐚 ‖^{2}$ two ways and comparing it with the law of cosines

differentiating the dot product with respect to the angle

The answer is: expanding $‖ 𝐛 - 𝐚 ‖^{2}$ two ways and comparing it with the law of cosines

assuming the vectors are unit vectors

The answer is: expanding $‖ 𝐛 - 𝐚 ‖^{2}$ two ways and comparing it with the law of cosines

9.[3p]
Match each expression to what it computes.

$𝐚 \cdot 𝐚$
$𝐯 / ‖ 𝐯 ‖$
$𝐛 - 𝐚$
$𝐅 \cdot 𝐝$

a unit vector along $𝐯$
the squared length of $𝐚$
the arrow from the tip of $𝐚$ to the tip of $𝐛$
the work done by a constant force

Show the answer

$𝐚 \cdot 𝐚$ : the squared length of $𝐚$ $𝐯 / ‖ 𝐯 ‖$ : a unit vector along $𝐯$ $𝐛 - 𝐚$ : the arrow from the tip of $𝐚$ to the tip of $𝐛$ $𝐅 \cdot 𝐝$ : the work done by a constant force

1.[1p] What is the length of the vector (2,3,6)?

2.[1p] What is (1,2)⋅(3,1)?

3.[2p] Two non-zero vectors are perpendicular exactly when

4.[2p] What is the angle, in degrees, between (1,2) and (3,1)?

5.[1p] A negative dot product means the angle between the two vectors is obtuse.

6.[2p] Which of these are operations that the definition of a vector space provides?

7.[3p] Projecting 𝐛=(4,3) onto 𝐚=(2,1) gives c𝐚. What is c?

8.[2p] The formula 𝐚⋅𝐛=‖𝐚‖‖𝐛‖cosθ is obtained by

9.[3p] Match each expression to what it computes.

1.[1p]
What is the length of the vector $(2, 3, 6)$ ?

2.[1p]
What is $(1, 2) \cdot (3, 1)$ ?

3.[2p]
Two non-zero vectors are perpendicular exactly when

4.[2p]
What is the angle, in degrees, between $(1, 2)$ and $(3, 1)$ ?

5.[1p]
A negative dot product means the angle between the two vectors is obtuse.

6.[2p]
Which of these are operations that the definition of a vector space provides?

7.[3p]
Projecting $𝐛 = (4, 3)$ onto $𝐚 = (2, 1)$ gives $c 𝐚$ . What is $c$ ?

8.[2p]
The formula $𝐚 \cdot 𝐛 = ‖ 𝐚 ‖ ‖ 𝐛 ‖ \cos θ$ is obtained by

9.[3p]
Match each expression to what it computes.