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 $\|\mathbf{b} - \mathbf{a}\|^2$ two ways and comparing it with the law of cosines

differentiating the dot product with respect to the angle

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

assuming the vectors are unit vectors

The answer is: expanding $\|\mathbf{b} - \mathbf{a}\|^2$ 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.