Q: If two planes have normal vectors $$\mathbf{n}_1$$ and $$\mathbf{n}_2$$, what is the direction vector of their line of intersection?

$$\mathbf{n}_1 \times \mathbf{n}_2$$

Q: A real-world application: An engineer needs to find the shortest distance from a point $$S(1,1,5) to a $$pipeline represented by the line $$x = 1 + t, \ y = 3 - t, \ z = 2t. $$What is the correct formula to use?

$$d = \frac{\left| \mathbf{PS} \times \mathbf{v} \right|}{|\mathbf{v}|}$$, \ where \ $$\mathbf{v}$$ \ is \ the \ direction \ vector \ of \ the \ line$

Question 1

Which of the following is the correct parametric equation for the line passing through the point $$P_0$$($$x_0$$, $$y_0$$, $$z_0$$)$$ and parallel to the vector $$\mathbf{v} = $$v_1$$ \mathbf{i} + $$v_2$$ \mathbf{j} + $$v_3$$ \mathbf{k}$$?

Accepted Answer

$$x = x_0$ + t v_1$, \ y = y_0$ + t v_2$, \ z = z_0$ + t v_3$

Question 2

Given the vector equation for a line $$\mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{v}$$, what does $$\mathbf{r}_0$$ represent?

Accepted Answer

The position vector of a point on the line

Question 3

A plane passes through the point $$P_0$$($$x_0$$, $$y_0$$, $$z_0$$)$$ and is normal to $$\mathbf{n} = A\mathbf{i} + B\mathbf{j} + C\mathbf{k}$$. Which of the following is the correct component equation for the plane?

Accepted Answer

$$A(x - x_0$) + B(y - y_0$) + C(z - z_0$) = 0$$

Question 4

What is the distance from a point $$S to a $$line through $$P$$ parallel to $$\mathbf{v}$$, given by the formula $$d = \frac{\left| \mathbf{PS} 	imes \mathbf{v} ight|}{|\mathbf{v}|}$$? What does $$\mathbf{PS}$$ represent?

Accepted Answer

The vector from $$P to$$ $$S$$

Question 5

Given two points $$P(-3,2,-3)$$ and $$Q(1,-1,4)$$, what is the parametric equation for the line segment joining $$P$$ and $$Q$$?

Accepted Answer

$$x = -3 + 4t, \ y = 2 - 3t, \ z = -3 + 7t, \ 0 \leq t \leq 1$$

Question 6

Which of the following best describes the geometric meaning of the cross product $$\mathbf{PS} 	imes \mathbf{v} in $$the context of finding the distance from a point to a line?

Accepted Answer

It gives a vector perpendicular to both $$\mathbf{PS}$$ and $$\mathbf{v}$$ whose magnitude is the area of the parallelogram formed by $$\mathbf{PS}$$ and $$\mathbf{v}$$

Question 7

A plane in space is given by $$3x + 2y + 6z = 6. $$What is a normal vector to this plane?

Accepted Answer

$$\mathbf{n} = 3\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}$$

Question 8

If two planes have normal vectors $$\mathbf{n}_1$$ and $$\mathbf{n}_2$$, what is the direction vector of their line of intersection?

Accepted Answer

$$\mathbf{n}_1 	imes \mathbf{n}_2$$

Question 9

Which of the following is the simplified component equation for a plane through $$P_0$$($$x_0$$, $$y_0$$, $$z_0$$)$$ with normal vector $$\mathbf{n} = A\mathbf{i} + B\mathbf{j} + C\mathbf{k}$$?

Accepted Answer

$$Ax + By + Cz = D$$, where $$D = Ax_0$ + By_0$ + Cz_0$

Question 10

A real-world application: An engineer needs to find the shortest distance from a point $$S(1,1,5) to a $$pipeline represented by the line $$x = 1 + t, \ y = 3 - t, \ z = 2t. $$What is the correct formula to use?

Accepted Answer

$$d = \frac{\left| \mathbf{PS} 	imes \mathbf{v} ight|}{|\mathbf{v}|}$$, \ where \ $$\mathbf{v}$$ \ is \ the \ direction \ vector \ of \ the \ line$

Question 11

Which of the following statements is true about the angle between two planes?

Accepted Answer

It is the same as the angle between their normal vectors

Question 12

Given the line $$x = -2 + 2t, \ y = 4t, \ z = 4 - 2t$$, what is the direction vector of the line?

Accepted Answer

$$\mathbf{v} = 2\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}$$

Question 13

A plane passes through $$(1,2,3)$$ and has normal vector $$\mathbf{n} = 2\mathbf{i} - \mathbf{j} + 4\mathbf{k}. $$What is the equation of the plane?

Accepted Answer

$$2(x-1) - (y-2) + 4(z-3) = 0$$

Question 14

If a line is given by $$\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}$$, which of the following statements is true?

Accepted Answer

The line passes through $$\mathbf{r}_0$$ and is parallel to $$\mathbf{v}$$

Question 15

A surveyor is mapping a property and needs to find the equation of the plane passing through $$(2, -1, 4)$$ and normal to $$\mathbf{n} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k}. $$What is the equation of the plane?

Accepted Answer

$$3(x-2) - 2(y+1) + (z-4) = 0$$

Question 16

Which of the following is the correct vector equation for a line through $$P_0(1$$,2,3)$$ parallel to $$\mathbf{v} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}$$?

Accepted Answer

$$\mathbf{r}(t) = \langle 1,2,3 \rangle + t \langle 2,3,-1 \rangle$$

Question 17

If the angle between two planes is $$	heta$$, how can it be found using their normal vectors $$\mathbf{n}_1$$ and $$\mathbf{n}_2$$?

Accepted Answer

$$\cos 	heta = \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{|\mathbf{n}_1||\mathbf{n}_2|}$$

Lines and Planes in Space: Parametric and Vector Equations, Distances, and Intersections

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