Vectors Cumulative Review
Chapter
Chapter 9
Section
Vectors Cumulative Review
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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
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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}.

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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}.

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0.48mins
Q1c

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

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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.

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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}|

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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}|

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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})|

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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})

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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})
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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.

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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}

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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}

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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}

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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})

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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).

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0.59mins
Q7

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

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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.

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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.

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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}.

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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.

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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.

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2.28mins
Q12

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

x + 2y + 2z - 6 = 0

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0.46mins
Q13a

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

2x - 3y = 0

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0.59mins
Q13b

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

3x - 2y + z= 0

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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'.

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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.

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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).

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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).

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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)

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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)

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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.

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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.

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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.

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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.

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1.18mins
Q19a

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

using coordinate axes you construct yourself, sketch this plane.

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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.

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1.56mins
Q20a

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

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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.

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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.

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2.33mins
Q21

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

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

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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}

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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}.

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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}.

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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.

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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.

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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}.

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0.50mins
Q24c

Solve the following system of equation:

  1. x-y+z=2

  2. -x+y+2z=1

  3. x-y+4z=5

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1.23mins
Q25a

Solve the following system of equation:

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

  2. x+2y+z=2

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

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1.45mins
Q25b

Solve the following system of equation:

  1. 2x-y+z=-1

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

  3. 2x+y-z=5

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1.40mins
Q25c

Solve the following system of equation:

  1. x-y-3z=1

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

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

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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

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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

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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

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1.27mins
Q26c

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

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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})

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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}.

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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.

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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'.

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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?

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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?

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0.37mins
Q31b

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

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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?

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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.

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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.)

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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.

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Q35

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.

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Q36