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

• the angle between the two vectors

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}.

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}.

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

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

b) Determine the angle between the two planes.

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}|

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}|

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})|

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})

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})

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

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}

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}

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}

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})

Determine the coordinates of the unit vector that is perpendicular to

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

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

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

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

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.

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}.

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.

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.

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

x + 2y + 2z - 6 = 0

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

2x - 3y = 0

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

3x - 2y + z= 0

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'.

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.

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).

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).

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)

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)

Determine a Cartesian equation for each of the following planes:

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

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.

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.

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

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

using coordinate axes you construct yourself, sketch this plane.

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.

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

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

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

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

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

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}

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

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

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

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

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.

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.

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}.

Solve the following system of equation:

1. x-y+z=2

2. -x+y+2z=1

3. x-y+4z=5

Solve the following system of equation:

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

2. x+2y+z=2

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

Solve the following system of equation:

1. 2x-y+z=-1

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

3. 2x+y-z=5

Solve the following system of equation:

1. x-y-3z=1

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

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

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

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

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

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

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})

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}.

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.

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'.

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?

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?

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

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?

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.

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.)

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.