Vectors Cumulative Review
Chapter
Chapter 9
Section
Vectors Cumulative Review
Solutions 67 Videos

For the vectors \vec{a}=(2,-1,-2) and \vec{b}=(3,-4,12), determine the following:

• the angle between the two vectors
1.01mins
Q1a

For the vectors \vec{a}=(2,-1,-2) and \vec{b}=(3,-4,12), determine the following:

the scalar and vector projections of \vec{a} on \vec{b}.

1.22mins
Q1b

For the vectors \vec{a}=(2,-1,-2) and \vec{b}=(3,-4, 12), determine the following:

the scalar and vector projections of \vec{b} on \vec{a}.

0.48mins
Q1c

Determine the line of intersection between \pi_1: 4x + 2y + 6z - 14 = 0 and \pi_2: x - y + z -5 = 0.

2.31mins
Q2a

\pi_1: 4x + 2y + 6z - 14 = 0 and \pi_2: x - y + z -5 = 0

b) Determine the angle between the two planes.

1.01mins
Q2b

If \vec{x} and \vec{y} are unit vectors, and the angle between them is 60^o, determine the value of each of the following:

|\vec{x}\cdot \vec{y}|

0.24mins
Q3a

If \vec{x} and \vec{y} are unit vectors, and the angle between them is 60^o, determine the value of each of the following:

|2\vec{x}\cdot 3\vec{y}|

0.19mins
Q3b

If \vec{x} and \vec{y} are unit vectors, and the angle between them is 60^o, determine the value of each of the following:

|(2\vec{x} - \vec{y})\cdot (\vec{x} + 3\vec{y})|

1.10mins
Q3c

Expand and simplify each of the following, where \vec{i}, \vec{j} and \vec{k} represent the standard basis vectors in \mathbb{R}^3

2(\vec{i} - 4\vec{j} + 3\vec{k})-4(2\vec{i} + 4\vec{j} +5\vec{k})-(\vec{i}-\vec{j})

0.56mins
Q4a

Expand and simplify each of the following, where \vec{i}, \vec{j} and \vec{k} represent the standard basis vectors in \mathbb{R}^3

• -2(3\vec{i} -4\vec{j} -5\vec{k})\cdot(2\vec{i} +3\vec{k}) + 2\vec{i} \cdot(3\vec{j}-2\vec{k})
0.45mins
Q4b

Determine the angle that the vector \vec{a} = (4, -2, -3) makes with the positive x-axis, y-axis, and z-axis.

1.10mins
Q5

If \vec{a} = (1, -2, 3), \vec{b} = (-1, 1, 2), and \vec{c} = (3, -4, -1), determine each of the following:

\vec{a} \times \vec{b}

0.32mins
Q6a

If \vec{a} = (1, -2, 3), \vec{b} = (-1, 1, 2), and \vec{c} = (3, -4, -1), determine each of the following:

2\vec{a} \times 3\vec{b}

0.23mins
Q6b

If \vec{a} = (1, -2, 3), \vec{b} = (-1, 1, 2), and \vec{c} = (3, -4, -1), determine

the area of the parallelogram determine by \vec{a} and \vec{b}

0.22mins
Q6c

If \vec{a} = (1, -2, 3), \vec{b} = (-1, 1, 2), and \vec{c} = (3, -4, -1), determine each of the following:

\vec{c} \cdot(\vec{b} \times \vec{a})

0.34mins
Q6d

Determine the coordinates of the unit vector that is perpendicular to

\vec{a} = (1, -1, 1) and \vec{b} = (2, -2, 3).

0.59mins
Q7

Determine vector and parametric equations for the line that contains A(2,-3,1) and B(1,2,3).

0.44mins
Q8a

A(2,-3,1) and B(1,2,3).

Verify that C(4, -13, -3) is on the line that contains A and B.

0.42mins
Q8b

Show that the lines L_1: \vec{r} = (2, 0, 9) + t(-1, 5 ,2), t\in \mathbb{R} and L_2: x - 3=\displaystyle{\frac{y + 5}{-5}} = \displaystyle{\frac{z -10}{-2}} are parallel and distinct.

1.34mins
Q9

Determine vector and parametric equations for the line that passes through (0, 0, 4) and its parallel to the line with parametric equations x = 1, y = 2+ t, and z = -3 + t, t \in \mathbb{R}.

0.50mins
Q10

Determine value of c such that the plane with equation 2x + 3y + cz - 8 = 0 is parallel to the line with equation \displaystyle{\frac{x - 1}{2}}=\displaystyle{\frac{y -2}{3}} =z + 1.

0.55mins
Q11

Determine the intersection of the line \displaystyle{\frac{x -2}{3}}= y+ 5=\displaystyle{\frac{z - 3}{5}} with the plane 5x + y - 2z + 2 = 0.

2.28mins
Q12

Sketch the following planes, and give two direction vectors for each.

x + 2y + 2z - 6 = 0

0.46mins
Q13a

Sketch the following planes, and give two direction vectors for each.

2x - 3y = 0

0.59mins
Q13b

Sketch the following planes, and give two direction vectors for each.

3x - 2y + z= 0

1.20mins
Q13c

If P(1, -2, 4) is reflected in the plane with equation 2x - 3y - 4z + 66 = 0, determine the coordinates of its image point, P'.

4.48mins
Q14

Determine the equation of the line that passes through the point A(1, 0, 2) and intersects the line \vec{r} = (-2, 3, 4) + s(1, 1, 2), s\in \mathbb{R}, at a right angle.

2.30mins
Q15

Determine the scalar equation of the plane that passes through the points A(1, 2, 3) B(-2, 0, 0), and C(1, 4, 0).

1.38mins
Q16a

Determine the distance from O(0, 0, 0) to the plane that is defined by A(1, 2, 3), B(-2, 0, 0), and C(1, 4, 0).

0.29mins
Q16b

Determine a Cartesian equation for each of the following plane:

the plane through the point A(-1, 2, 5) with \vec{n} = (3, -5, 4)

0.36mins
Q17a

Determine a Cartesian equation for each of the following plane:

the plane through the point K(4, 1, 2) and perpendicular to the line joining the points (2, 1, 8) and (1, 2, -4)

1.59mins
Q17b

Determine a Cartesian equation for each of the following planes:

the plane though the point (3, -1, 3) and perpendicular to the z-axis.

0.44mins
Q17c

Determine a Cartesian equation for each of the following planes:

the plane through the point (3, 1, -2) and (1, 3, -1) and parallel to the y-axis.

1.41mins
Q17d

An airplane heads due north with a velocity of 400 km/h and encounters a wind of 100 km/h from the northeast. Determine the resultant velocity of the airplane.

2.07mins
Q18

Determine a vector equation for the plane with Cartesian equation 3x -2y + z - 6 = 0, and verify that your vector equation is correct.

1.18mins
Q19a

For 3x -2y + z - 6 = 0,

using coordinate axes you construct yourself, sketch this plane.

0.40mins
Q19b

A line with equation \vec{r} = (1,0, -2) + s(2, -1, 2), s\in \mathbb{R}, intersects the plane x + 2y + z = 2 at an angle of \theta degrees. Determine this angle to the nearest degree.

1.56mins
Q20a

Show that the planes \pi_1: 2x - 3y + z -1 = 0 and \pi_2: 4x - 3y - 17z= 0 are perpendicular.

0.53mins
Q20b

Show that the planes \pi_3: 2x -3y + 2z - 1= 0 and \pi_4: 2x - 3y + 2z -3 = 0 are parallel but not coincident.

1.18mins
Q20c

Two forces, 25N and 40N, have an angle of 60^o between them . Determine the resultant and equilibriant of these two vectors.

2.33mins
Q21

You are given the vectors \vec{a} and \vec{b}, as shown on the bottom.

Sketch \vec{a} -\vec{b}

0.44mins
Q22a

You are given the vectors \vec{a} and \vec{b}, as shown on the bottom.

Sketch 2\vec{a} + \displaystyle{\frac{1}{2}}\vec{b}

0.30mins
Q22b

If \vec{a} = (6 ,2, -3) determine the following:

the coordinates of a unit vector in the same direction as \vec{a}.

0.31mins
Q23a

If \vec{a} = (6 ,2, -3) determine the following:

the coordinates of a unit vector in the opposite direction as \vec{a}.

0.15mins
Q23b

A parallelogram OBCD has one vertex at O(0, 0) and two of its remaining three vertices at B(-1, 7) and D(9, 2).

Determine a vector that is equivalent to each of the two diagonals.

2.38mins
Q24a

A parallelogram OBCD has one vertex at O(0, 0) and two of its remaining three vertices at B(-1, 7) and D(9, 2).

Determine the angel between these diagonals.

0.59mins
Q24b

A parallelogram OBCD has one vertex at O(0, 0) and two of its remaining three vertices at B(-1, 7) and D(9, 2).

Determine the angle between \vec{OB} and \vec{OD}.

0.50mins
Q24c

Solve the following system of equation:

1. x-y+z=2

2. -x+y+2z=1

3. x-y+4z=5

1.23mins
Q25a

Solve the following system of equation:

1. -2x-3y+z=-11

2. x+2y+z=2

3. -x-y+3z=-12

1.45mins
Q25b

Solve the following system of equation:

1. 2x-y+z=-1

2. 4x-2y+2z=-2

3. 2x+y-z=5

1.40mins
Q25c

Solve the following system of equation:

1. x-y-3z=1

2. 2x-2y-6z=2

3. -4x+4y+12z=-4

0.39mins
Q25d

State whether each of the following pairs of planes intersect. If the planes do intersect,determine the equation of their line of intersection.

1. x-y+z-1=0

2. x+2y-2z+2=0

1.20mins
Q26a

State whether each of the following pairs of planes intersect. If the planes do intersect,determine the equation of their line of intersection.

1. x-4y+7z=28

2. 2x-8y+14z=60

0.22mins
Q26b

State whether each of the following pairs of planes intersect. If the planes do intersect,determine the equation of their line of intersection.

1. x-y+z-2=0

2. 2x+y+z-4=0

1.27mins
Q26c

Determine the angle between the line with symmetric equations x=-y, z=4 and the plane 2x-2z=5.

1.26mins
Q27

If \vec{a} and \vec{b} are unit vectors, and the angle between them is 60^{\circ}, calculate (6\vec{a}+\vec{b})\cdot(\vec{a}-2\vec{b})

1.01mins
Q28a

Calculate the dot product of 4\vec{x}=\vec{y} and 2\vec{x}+3\vec{y} if |\vec{x}|=3, and the angle between \vec{x} and \vec{y} is 60^{\circ}.

1.22mins
Q28b

A line that passes through the origin is perpendicular to a plane \pi and intersects the plane at (-1,3,1). Determine an equation for this line and the Cartesian equation of the plane.

1.30mins
Q29

The point P(-1,0,1) is reflected in the plane \pi:y-z=0 and has P' as its image. Determine the coordinates of the point P'.

2.34mins
Q30

A river is 2 km wide and flows at 4 km/h. A motorboat that has a speed of 10 km/h in still water heads out from one bank, which is perpendicular to the current. A marina lines directly across the river, on the opposite bank.

How far downstream form the marina will the motorboat touch the other bank?

1.32mins
Q31a

A river is 2 km wide and flows at 4 km/h. A motorboat that has a speed of 10 km/h in still water heads out from one bank, which is perpendicular to the current. A marina lines directly across the river, on the opposite bank.

How long will it take for the motorboat to reach the other bank?

0.37mins
Q31b

Determine the equation of the line passing through A(2,-1,3) and B(6,3,4).

0.37mins
Q32a

Does the line which passes through A(2,-1,3) and B(6,3,4) lie on the plane with equation x-2y+4z-16=0?

0.58mins
Q32b

A sailboat is acted upon by a water current and the wind. the velocity of the wind is 16 km/h from the west, and the velocity of the current is 12 km/h from the south. Find the resultant of these two velocities.

1.34mins
Q33

A crate has a mass of 400 kg and is sitting on an inclined plane that makes an angle of 30° with the level ground. Determine the components of the weight of the mass, perpendicular and parallel to the plane. (Assume that a 1 kg mass exerts a force of 9.8 N.)

Q34

State whether each of the following is true or false. Justify your answer.

a) Any two non-parallel lines in R^2 must always intersect at a point.

b) Any two non-parallel planes in R^3 must always intersect on a line.

c) The line with equation x =y = z will always intersect the plane with equation x - 2y + 2z = k, regardless of the value of k.

d) The lines \frac{x}{2} = y = 1 = \frac{z + 1}{3} and \frac{x -1}{-4} = \frac{y - 1}{-2} =\frac{z + 1}{-2} are parallel.

Consider the lines L_1: x =2, \frac{y-2}{3}= z and L_2: x = y + k = \frac{z + 14}{k}.
a) Explain why the lines can never be parallel, regardless of the value of k.
b) Determine the value of k that makes these two lines intersect at a single point, and find the actual point of intersection.