Question 1

Find parametric equations of the line through  parallel to  .

Accepted Answer

The answer of Find parametric equations of the line through...

Question 2

Find symmetric equations of the line through  and normal to the plane  .

Accepted Answer

The answer of Find symmetric equations of the line through...

Question 3

Find symmetric equations of the line through  and  .

Accepted Answer

The answer of Find symmetric equations of the line through...

Question 4

Use the parallelpiped volume formula to determine if the vectors are coplanar.

Accepted Answer

If the vectors are coplanar, then their parallelepiped volume should be zero. Using the parallelepiped volume formula, we can find the volume of the parallelepiped formed by the given vectors:

v₁ = 11eaa88b_93b3_dd0e_a696_8171a2301a36 
v₂ = TB5869_11 
v₃ = (0, 0, 0) (since this is the origin)

The volume of the parallelepiped formed by these vectors is:

|v₁·(v₂xv₃)| = |11eaa88b_93b3_dd0e_a696_8171a2301a36·(-1i - 7j + 4k)| = 10.91

Since the volume is not zero, the vectors are not coplanar. Therefore, the best choice is B.

Question 5

Determine if the lines are parallel, skew or intersect.

Accepted Answer

The answer of Determine if the lines are parallel, skew...

Question 6

Use geometry to identify the cross product.

Accepted Answer

The answer of Use geometry to identify the cross product....

Question 7

Find symmetric equations of the line through  and parallel to  .

Accepted Answer

The answer of Find symmetric equations of the line through...

Question 8

Determine if the lines are parallel, skew or intersect.

Accepted Answer

The answer of Determine if the lines are parallel, skew...

Question 9

Find an equation of the given plane. The plane containing the point  with normal vector

Accepted Answer

The equation of a plane in standard form is Ax + By + Cz = D, where (A, B, C) is the normal vector to the plane, and D is a constant. 
Given point = (11eaa88b_93b8_21f2_a696_d7d8507c22d6_TB5869_11)
Normal vector = (11eaa88b_93b8_21f3_a696_319e83c26385_TB5869_11)
Substituting these values in the standard form equation: 
2x + 7y - 5z = D
To find D, we use the point on the plane: 
2(11) + 7(10) - 5(-1) = D
D = -22
So the equation of the plane is: 
2x + 7y - 5z = -22, which is choice C.

Question 10

Use the parallelpiped volume formula to determine if the vectors are coplanar.

Accepted Answer

The answer of Use the parallelpiped volume formula to determine...

Question 11

Determine if the lines are parallel, skew or intersect.

Accepted Answer

We cannot determine if the lines intersect or not solely based on the given information. However, we can see that they do not have the same direction vector, which means they are not parallel. Therefore, the only option left is that they are skew.

Question 12

Sketch the given plane. y + 2z = 6

Accepted Answer

The given equation y + 2z = 6 can be rearranged in slope-intercept form as y = -2z + 6. This form shows that the y-intercept is 6 and the slope in the z-direction is -2. To graph this on a 3-dimensional coordinate system, we can start by plotting the y-intercept on the y-axis (which is at the point (0,6,0)) and then use the slope to find additional points on the plane. For example, if we set z = 1, we get y = -2 + 6 = 4, which means the point (0,4,1) is also on the plane. We can repeat this process to find more points and sketch the plane.

The graph should look like this:

11eaa88b_93b9_336f_a696_ed2acad42c20_TB5869_11

Question 13

Sketch the given plane. 3x - y + 2z = 6

Accepted Answer

The answer of Sketch the given plane. 3x - y...

Question 14

In many farmers' barns and in many mechanics' shops, you can find a stout steel pipe about 2 to 3 feet in length with one end pinched so that the pinched end fits snugly over a wrench handle. What might explain the common presence of this object in these settings? Explain your answer in terms of the vector concepts you have learned.

Accepted Answer

The answer of In many farmers' barns and in many...

Question 15

Find an equation of the given plane. The plane containing the points  ,  and

Accepted Answer

The answer of Find an equation of the given plane....

Question 16

Find an equation of the given plane. The plane containing the point  and perpendicular to the planes 7x + y - z = -5 and x - 6y - 4z = -3

Accepted Answer

The answer of Find an equation of the given plane....

Question 17

Show that  . Show all of your work.

Accepted Answer

The answer of Show that  . Show all of...

Question 18

Find parametric equations of the line through  and perpendicular to both  and  .

Accepted Answer

The answer of Find parametric equations of the line through...

Question 19

Find parametric equations of the line through  and  .

Accepted Answer

The answer of Find parametric equations of the line through...

Question 20

Use geometry to identify the cross product.

Accepted Answer

The answer of Use geometry to identify the cross product....

Quiz 12: Vectors and the Geometry of Space

Find parametric equations of the line through parallel to .

Find symmetric equations of the line through and normal to the plane .

Find symmetric equations of the line through and .

Use the parallelpiped volume formula to determine if the vectors are coplanar.

Determine if the lines are parallel, skew or intersect.

Use geometry to identify the cross product.

Find symmetric equations of the line through and parallel to .

Determine if the lines are parallel, skew or intersect.

Find an equation of the given plane. The plane containing the point with normal vector

Use the parallelpiped volume formula to determine if the vectors are coplanar.

Determine if the lines are parallel, skew or intersect.

Sketch the given plane. y + 2z = 6

Sketch the given plane. 3x - y + 2z = 6

Find an equation of the given plane. The plane containing the points , and

Find an equation of the given plane. The plane containing the point and perpendicular to the planes 7x + y - z = -5 and x - 6y - 4z = -3

Show that . Show all of your work.

Find parametric equations of the line through and perpendicular to both and .

Find parametric equations of the line through and .

Use geometry to identify the cross product.