Lesson test: Space and vectors

Space and vectors

1.[1p]
Find the distance between the points $(2, - 1, 5)$ and $(8, 1, 8)$ .

CorrectNot quite: 7

2.[2p]
Find the angle between $⟨ 1, 2, 2 ⟩$ and $⟨ 2, - 1, 2 ⟩$ , in degrees, rounded to one decimal place.

CorrectNot quite: 63.6

3.[2p]
A constant force $𝐅 = ⟨ 3, - 2, 5 ⟩$ newtons moves an object from $(0, 1, 2)$ to $(4, 3, 6)$ , in metres. How many joules of work does it do?

CorrectNot quite: 28

4.[3p]
Find the area of the triangle with corners $(0, 0, 0)$ , $(2, 1, 0)$ and $(0, 3, 4)$ , rounded to two decimal places.

CorrectNot quite: 5.39

5.[2p]
A force $𝐅 = ⟨ 0, 40, 30 ⟩$ newtons is applied at $𝐫 = ⟨ 0.3, 0, 0 ⟩$ metres from a pivot. What is the magnitude of the torque $𝐫 \times 𝐅$ , in newton metres?

CorrectNot quite: 15

6.[1p]
What is $𝐢 \times 𝐤$ ?

$- 𝐣$

Correct

$𝐣$

The answer is: $-\mathbf{j}$

$𝟎$

The answer is: $-\mathbf{j}$

$𝐢 + 𝐤$

The answer is: $-\mathbf{j}$

7.[2p]
The line $x = 1 + 2 t$ , $y = - 1 + t$ , $z = 4 - 2 t$ meets the plane $x + y + z = 9$ at one point. What is the $x$ coordinate of that point?

CorrectNot quite: 11

8.[3p]
Find the distance from the point $(3, 0, 2)$ to the plane $2 x - 2 y + z = 1$ , rounded to two decimal places.

CorrectNot quite: 2.33

9.[3p]
Which of these statements are correct?

Select all that apply

$𝐛 \times 𝐚 = - 𝐚 \times 𝐛$ for all vectors $𝐚$ and $𝐛$

Correct

$𝐚 \times 𝐛$ is perpendicular to both $𝐚$ and $𝐛$

Correct

The plane $4 x - y + 2 z = 5$ has normal vector $⟨ 4, - 1, 2 ⟩$

Correct

$| 𝐚 \times 𝐛 | = | 𝐚 | | 𝐛 | \cos θ$

The answer is: $\mathbf{b} \times \mathbf{a} = -\mathbf{a} \times \mathbf{b}$ for all vectors $\mathbf{a}$ and $\mathbf{b}$, $\mathbf{a} \times \mathbf{b}$ is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$, The plane $4x - y + 2z = 5$ has normal vector $\langle 4, -1, 2 \rangle$

Two lines in space that are not parallel must meet

1.[1p] Find the distance between the points (2,-1,5) and (8,1,8).

2.[2p] Find the angle between ⟨1,2,2⟩ and ⟨2,-1,2⟩, in degrees, rounded to one decimal place.

3.[2p] A constant force 𝐅=⟨3,-2,5⟩ newtons moves an object from (0,1,2) to (4,3,6), in metres. How many joules of work does it do?

4.[3p] Find the area of the triangle with corners (0,0,0), (2,1,0) and (0,3,4), rounded to two decimal places.

5.[2p] A force 𝐅=⟨0,40,30⟩ newtons is applied at 𝐫=⟨0.3,0,0⟩ metres from a pivot. What is the magnitude of the torque 𝐫×𝐅, in newton metres?

6.[1p] What is 𝐢×𝐤?

7.[2p] The line x=1+2t, y=-1+t, z=4-2t meets the plane x+y+z=9 at one point. What is the x coordinate of that point?

8.[3p] Find the distance from the point (3,0,2) to the plane 2x-2y+z=1, rounded to two decimal places.

9.[3p] Which of these statements are correct?

1.[1p]
Find the distance between the points $(2, - 1, 5)$ and $(8, 1, 8)$ .

2.[2p]
Find the angle between $⟨ 1, 2, 2 ⟩$ and $⟨ 2, - 1, 2 ⟩$ , in degrees, rounded to one decimal place.

3.[2p]
A constant force $𝐅 = ⟨ 3, - 2, 5 ⟩$ newtons moves an object from $(0, 1, 2)$ to $(4, 3, 6)$ , in metres. How many joules of work does it do?

4.[3p]
Find the area of the triangle with corners $(0, 0, 0)$ , $(2, 1, 0)$ and $(0, 3, 4)$ , rounded to two decimal places.

5.[2p]
A force $𝐅 = ⟨ 0, 40, 30 ⟩$ newtons is applied at $𝐫 = ⟨ 0.3, 0, 0 ⟩$ metres from a pivot. What is the magnitude of the torque $𝐫 \times 𝐅$ , in newton metres?

6.[1p]
What is $𝐢 \times 𝐤$ ?

7.[2p]
The line $x = 1 + 2 t$ , $y = - 1 + t$ , $z = 4 - 2 t$ meets the plane $x + y + z = 9$ at one point. What is the $x$ coordinate of that point?

8.[3p]
Find the distance from the point $(3, 0, 2)$ to the plane $2 x - 2 y + z = 1$ , rounded to two decimal places.

9.[3p]
Which of these statements are correct?