equation of line in space, 20 numerical mcqs in it with explained answers
Here is a set of 20 numerical multiple-choice questions (MCQs) on the topic "Equation of the Line in Space," along with detailed explanations for each answer. This covers parametric, symmetric, and vector forms of lines, as well as their geometric interpretations.
### Numerical MCQs with Explained Answers: Equation of Line in Space
1. **Find the parametric equations of a line passing through point $$ (1, 2, 3) $$ and parallel to the vector $$ \vec{d} = (4, -2, 5) $$.**
A) $$x=1+4t, y=2-2t, z=3+5t$$
B) $$x=4+t, y=2t-2, z=5+3t$$
C) $$x=t, y=2t, z=3t$$
D) $$x=1-4t, y=2+2t, z=3-5t$$
**Answer:** A) $$x=1+4t, y=2-2t, z=3+5t$$.
**Explanation:** Parametric form is given by $$x=x_0+at, y=y_0+bt, z=z_0+ct$$ where $$\vec{d} = (a,b,c)$$ is the direction vector and $$(x_0, y_0, z_0)$$ is a point on the line.
2. **Find the symmetric form of the line through $$ (2, -1, 5) $$ with direction vector $$ (3, 4, -2) $$.**
A) $$\frac{x-2}{3} = \frac{y+1}{4} = \frac{z-5}{-2}$$
B) $$\frac{x+2}{3} = \frac{y-1}{4} = \frac{z+5}{-2}$$
C) $$\frac{x-2}{-3} = \frac{y+1}{-4} = \frac{z-5}{2}$$
D) $$\frac{x+2}{-3} = \frac{y-1}{-4} = \frac{z+5}{2}$$
**Answer:** A) $$\frac{x-2}{3} = \frac{y+1}{4} = \frac{z-5}{-2}$$.
**Explanation:** Symmetric form is $$\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$$.
3. **If the vector form of a line is $$\vec{r} = \begin{pmatrix}1 \\ -3 \\ 2\end{pmatrix} + t \begin{pmatrix}4 \\ 0 \\ -1\end{pmatrix}$$, what is the point and direction vector?**
A) Point (1, -3, 2), direction (4, 0, -1)
B) Point (4, 0, -1), direction (1, -3, 2)
C) Point (0, 0, 0), direction (4, 0, -1)
D) Point (1, 2, -3), direction (-4, 0, 1)
**Answer:** A) Point (1, -3, 2), direction (4, 0, -1).
**Explanation:** Vector form is $$\vec{r} = \vec{r_0} + t\vec{d}$$, $$\vec{r_0}$$ is point and $$\vec{d}$$ is direction.
4. **What is the equation of the line passing through $$ (0, 0, 0) $$ and perpendicular to the plane $$ 2x + 3y - z = 6 $$?**
A) $$ \frac{x}{2} = \frac{y}{3} = \frac{z}{-1} $$
B) $$ \frac{x}{-2} = \frac{y}{-3} = \frac{z}{1} $$
C) $$ \frac{x}{2} = \frac{y}{-3} = \frac{z}{1} $$
D) $$ \frac{x}{3} = \frac{y}{2} = \frac{z}{1} $$
**Answer:** A) $$ \frac{x}{2} = \frac{y}{3} = \frac{z}{-1} $$.
**Explanation:** Direction vector is the normal to the plane $$\vec{n} = (2, 3, -1)$$.
5. **Find the distance between the point $$ (3, -2, 1) $$ and the line $$ \frac{x-1}{2} = \frac{y+1}{-3} = \frac{z}{4} $$.**
A) 5
B) 4
C) 3
D) 2
**Answer:** D) 2.
**Explanation:** Distance = $$\frac{|(\vec{P_0P} \times \vec{d})|}{|\vec{d}|}$$ where $$\vec{P_0}$$ is point on line, $$\vec{P}$$ is given point.
6. **If two lines are given by $$ \frac{x-2}{1} = \frac{y}{-2} = \frac{z-1}{3} $$ and $$ \frac{x+1}{2} = \frac{y-2}{1} = \frac{z}{-1} $$, are they parallel?**
A) Yes
B) No
**Answer:** B) No.
**Explanation:** Direction vectors $$(1, -2, 3)$$ and $$(2, 1, -1)$$ are not scalar multiples.
7. **Find the vector form of the line passing through $$ (1, 0, -1) $$ and parallel to the line $$ \frac{x}{2} = \frac{y-3}{4} = \frac{z+2}{-1} $$.**
A) $$\vec{r} = \begin{pmatrix}1 \\ 0 \\ -1\end{pmatrix} + t \begin{pmatrix}2 \\ 4 \\ -1\end{pmatrix}$$
B) $$\vec{r} = \begin{pmatrix}2 \\ 4 \\ -1\end{pmatrix} + t \begin{pmatrix}1 \\ 0 \\ -1\end{pmatrix}$$
C) $$\vec{r} = \begin{pmatrix}1 \\ 0 \\ -1\end{pmatrix} + t \begin{pmatrix}1 \\ 2 \\ -4\end{pmatrix}$$
D) $$\vec{r} = t \begin{pmatrix}1 \\ 0 \\ -1\end{pmatrix} + \begin{pmatrix}2 \\ 4 \\ -1\end{pmatrix}$$
**Answer:** A) $$\vec{r} = \begin{pmatrix}1 \\ 0 \\ -1\end{pmatrix} + t \begin{pmatrix}2 \\ 4 \\ -1\end{pmatrix}$$.
**Explanation:** Use given point and direction vector from the line.
8. **A line passes through points $$ A(1, 2, 3) $$ and $$ B(4, 0, 1) $$. Find its parametric equation.**
A) $$x=1+3t, y=2-2t, z=3-2t$$
B) $$x=4+3t, y=0-2t, z=1-2t$$
C) $$x=1-t, y=2+t, z=3+t$$
D) $$x=1+4t, y=2+0t, z=3+1t$$
**Answer:** A) $$x=1+3t, y=2-2t, z=3-2t$$.
**Explanation:** Direction vector $$\vec{AB}=(4-1, 0-2, 1-3) = (3, -2, -2)$$.
9. **Find the symmetric form of the line through $$ (0, 0, 0) $$ parallel to $$ \vec{d} = (1, -1, 2) $$.**
A) $$\frac{x}{1} = \frac{y}{-1} = \frac{z}{2}$$
B) $$\frac{x-1}{1} = \frac{y+1}{-1} = \frac{z-2}{2}$$
C) $$\frac{x}{-1} = \frac{y}{1} = \frac{z}{-2}$$
D) $$\frac{x+1}{1} = \frac{y-1}{-1} = \frac{z+2}{2}$$
**Answer:** A) $$\frac{x}{1} = \frac{y}{-1} = \frac{z}{2}$$.
**Explanation:** Since line passes through origin, $$\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}$$ reduces to above.
10. **Find the Cartesian equation of the line passing through $$ (2, -1, 4) $$ and parallel to vector $$ \vec{d} = (1, 3, -2) $$.**
A) $$\frac{x-2}{1} = \frac{y+1}{3} = \frac{z-4}{-2}$$
B) $$\frac{x-2}{1} = \frac{y-1}{3} = \frac{z-4}{2}$$
C) $$\frac{x+2}{1} = \frac{y+1}{3} = \frac{z+4}{-2}$$
D) $$\frac{x-2}{-1} = \frac{y+1}{-3} = \frac{z-4}{2}$$
**Answer:** A) $$\frac{x-2}{1} = \frac{y+1}{3} = \frac{z-4}{-2}$$.
**Explanation:** Direct substitution into symmetric form.
11. **Find the parameter $$ t $$ at which line $$ \vec{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix} $$ passes through $$ (9, 1, 7) $$.**
A) 2
B) 3
C) 1
D) 4
**Answer:** A) 2.
**Explanation:** Solve system $$1+4t=9, 2 - t = 1, 3 + 2t =7$$.
12. **Find the direction ratios of the line passing through $$ (1, 0, 2) $$ and making equal angles with coordinate axes.**
A) (1, 1, 1)
B) (1, -1, 1)
C) (0, 1, 1)
D) (1, 0, 1)
**Answer:** A) (1, 1, 1).
**Explanation:** Direction ratios are equal if angles with axes are equal.
13. **If the line has symmetric equations $$\frac{x-1}{2} = \frac{y+3}{-4} = \frac{z}{1}$$, find the parametric equations.**
A) $$x=1+2t, y=-3-4t, z=t$$
B) $$x=2t, y=-4t, z=1+t$$
C) $$x=1-2t, y=3+4t, z=-t$$
D) $$x=t, y=1 - 3t, z = 4t$$
**Answer:** A) $$x=1+2t, y=-3-4t, z=t$$.
**Explanation:** Parametric form directly from symmetric form.
14. **Find the point on line $$ \vec{r} = \begin{pmatrix} 5 \\ 1 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix} $$ when $$ t = -1 $$.**
A) (3, 4, -5)
B) (7, -2, 3)
C) (3, -2, 3)
D) (5, 1, -1)
**Answer:** A) (3, 4, -5).
**Explanation:** Substitute $$t = -1$$.
15. **Two lines intersect if the system derived from their parametric equations has:**
A) No solution
B) Infinite solutions
C) One unique solution
D) Depends on the situation
**Answer:** C) One unique solution.
**Explanation:** One unique $$t$$ and $$s$$ solving the system means intersection.
16. **The line $$ \vec{r} = \vec{a} + t \vec{b} $$ is coplanar with the plane which has equation $$ \vec{n} \cdot (\vec{r} - \vec{r_0})=0 $$ if:**
A) $$\vec{b} \cdot \vec{n} = 0$$
B) $$\vec{a} \cdot \vec{b} = 0$$
C) $$\vec{a} \cdot \vec{n} = 0$$
D) $$\vec{b} \neq 0$$
**Answer:** A) $$\vec{b} \cdot \vec{n} = 0$$.
**Explanation:** Line direction vector $$\vec{b}$$ is perpendicular to the plane normal $$\vec{n}$$.
17. **Calculate the shortest distance between lines $$ \frac{x}{1} = \frac{y-2}{2} = \frac{z+1}{3} $$ and $$ \frac{x-1}{2} = \frac{y}{-1} = \frac{z}{1} $$.**
A) $$\frac{|\vec{PQ} \cdot (\vec{d_1} \times \vec{d_2})|}{|\vec{d_1} \times \vec{d_2}|}$$
B) $$\frac{|\vec{PQ} \times (\vec{d_1} + \vec{d_2})|}{|\vec{d_1} + \vec{d_2}|}$$
C) $$0$$
D) Undefined
**Answer:** A) $$\frac{|\vec{PQ} \cdot (\vec{d_1} \times \vec{d_2})|}{|\vec{d_1} \times \vec{d_2}|}$$.
**Explanation:** Formula for shortest distance between skew lines.
18. **If a line has direction vector $$ (1, 2, 3) $$ and passes through $$ (1, 1, 1) $$, find the vector equation.**
A) $$ \vec{r} = \begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix} + t \begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix} $$
B) $$ \vec{r} = t \begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix} + \begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix} $$
C) $$ \vec{r} = \begin{pmatrix}0 \\ 0 \\ 0\end{pmatrix} + t \begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix} $$
D) $$ \vec{r} = \begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix} + t \begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix} $$
**Answer:** A) $$ \vec{r} = \begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix} + t \begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix} $$.
**Explanation:** Point plus parameter times direction vector.
19. **If the vector equation of a line is $$\vec{r} = \vec{a} + t \vec{b}$$, the line intersects the x-axis when:**
A) $$y=0$$, $$z=0$$
B) $$x=0$$
C) $$t=0$$
D) $$x=y=z$$
**Answer:** A) $$y=0$$, $$z=0$$.
**Explanation:** Points on x-axis have zero y and z coordinates.
20. **Find the angle between lines given by vectors $$\vec{d_1} = (1, 2, 2)$$ and $$\vec{d_2} = (2, 1, -2)$$.**
A) $$\cos^{-1}\frac{0}{\sqrt{9} \cdot \sqrt{9}}$$
B) $$\cos^{-1}\frac{2}{\sqrt{9} \cdot \sqrt{9}}$$
C) $$\cos^{-1}\frac{3}{\sqrt{9} \cdot \sqrt{9}}$$
D) $$\cos^{-1}\frac{-1}{\sqrt{9} \cdot \sqrt{9}}$$
**Answer:** D) $$\cos^{-1}\frac{-1}{3 \cdot 3} = \cos^{-1} \left(-\frac{1}{9}\right) $$.
**Explanation:** Angle formula $$\cos \theta = \frac{\vec{d_1} \cdot \vec{d_2}}{|\vec{d_1}||\vec{d_2}|}$$.
21 to 40
Here are numerical MCQs 21 to 40 on the topic "Equation of Line in Space" with explained answers:
21. **Find the coordinates of the point on the line $$ \vec{r} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix} + t \begin{pmatrix} -2 \\ 4 \\ 1 \end{pmatrix} $$ when $$ t = \frac{1}{2} $$.**
A) $$ (2, 1, 2.5) $$
B) $$ (4, 1, 3) $$
C) $$ (3, 4, 2.5) $$
D) $$ (2, -3, 2.5) $$
**Answer:** A) $$ (2, 1, 2.5) $$.
**Explanation:** Substitute $$t = \frac{1}{2}$$ into $$x = 3 - 2t$$, $$y = -1 + 4t$$, $$z = 2 + t$$.
22. **What is the parametric form of the line where $$x = 2t + 1$$, $$y = 3t - 4$$, and $$z = -t + 7$$?**
A) Passes through $$ (1, -4, 7) $$ and direction vector $$ (2, 3, -1) $$
B) Passes through $$ (0, 0, 0) $$ and direction vector $$ (2, 3, -1) $$
C) Passes through $$ (2, 3, -1) $$ and direction vector $$ (1, -4, 7) $$
D) Passes through $$ (1, 0, 0) $$ and direction vector $$ (2, 3, 1) $$
**Answer:** A) Passes through $$ (1, -4, 7) $$ and direction vector $$ (2, 3, -1) $$.
**Explanation:** Set $$t=0$$ to find point, coefficients of $$t$$ give direction vector.
23. **Line 1: $$\frac{x}{1} = \frac{y}{-2} = \frac{z}{3}$$, Line 2: $$\frac{x-1}{2} = \frac{y+1}{4} = \frac{z}{-6}$$. Are these lines parallel?**
A) Yes
B) No
**Answer:** B) No.
**Explanation:** Direction vectors are $$(1, -2, 3)$$ and $$(2, 4, -6)$$ which are not scalar multiples.
24. **Find the distance from point $$ (4, 2, -1) $$ to the line $$ \frac{x-1}{2} = \frac{y}{-1} = \frac{z+3}{1} $$.**
A) $$3$$
B) $$\frac{5}{\sqrt{6}}$$
C) $$2$$
D) $$\frac{4}{\sqrt{6}}$$
**Answer:** B) $$\frac{5}{\sqrt{6}}$$.
**Explanation:** Use distance formula involving cross product of vector $$ \vec{P_0P} $$ and direction vector.
25. **The line passing through points $$ (1, -3, 5) $$ and $$ (4, 0, 1) $$ can be represented by which vector equation?**
A) $$ \vec{r} = \begin{pmatrix}1 \\ -3 \\ 5 \end{pmatrix} + t \begin{pmatrix}3 \\ 3 \\ -4 \end{pmatrix} $$
B) $$ \vec{r} = \begin{pmatrix}4 \\ 0 \\ 1 \end{pmatrix} + t \begin{pmatrix}1 \\ -3 \\ 5 \end{pmatrix} $$
C) $$ \vec{r} = \begin{pmatrix}1 \\ -3 \\ 5 \end{pmatrix} + t \begin{pmatrix}4 \\ 0 \\ 1 \end{pmatrix} $$
D) $$ \vec{r} = t \begin{pmatrix}3 \\ 3 \\ -4 \end{pmatrix} + \begin{pmatrix}1 \\ -3 \\ 5 \end{pmatrix} $$
**Answer:** A) $$ \vec{r} = \begin{pmatrix}1 \\ -3 \\ 5 \end{pmatrix} + t \begin{pmatrix}3 \\ 3 \\ -4 \end{pmatrix} $$.
**Explanation:** Direction vector is $$B - A = (3, 3, -4)$$.
26. **Determine the parameter $$t$$ for which the line $$x = 1 - t$$, $$y = 3 + 2t$$, $$z = 4t - 5$$ passes through the point $$ (4, 11, 15) $$.**
A) 3
B) 4
C) 5
D) 6
**Answer:** C) 5.
**Explanation:** Solve $$1 - t = 4$$, $$3 + 2t = 11$$, $$4t - 5 = 15$$ for consistent $$t$$.
27. **Find the direction cosines of the vector $$ (2, -3, 6) $$.**
A) $$ \left(\frac{2}{7}, \frac{-3}{7}, \frac{6}{7}\right) $$
B) $$ \left(\frac{2}{\sqrt{49}}, \frac{-3}{\sqrt{49}}, \frac{6}{\sqrt{49}}\right) $$
C) $$ \left(\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\right) $$
D) $$ \left(\frac{2}{\sqrt{49}}, \frac{3}{\sqrt{49}}, \frac{6}{\sqrt{49}}\right) $$
**Answer:** B) $$ \left(\frac{2}{7}, \frac{-3}{7}, \frac{6}{7}\right) $$.
**Explanation:** Direction cosine = component divided by vector magnitude; magnitude $$= \sqrt{4+9+36} = 7$$.
28. **Find the symmetric equation of the line passing through points $$ (1, 2, 3) $$ and $$ (4, 5, 6) $$.**
A) $$\frac{x-1}{3} = \frac{y-2}{3} = \frac{z-3}{3}$$
B) $$\frac{x+1}{3} = \frac{y+2}{3} = \frac{z+3}{3}$$
C) $$\frac{x-4}{3} = \frac{y-5}{3} = \frac{z-6}{3}$$
D) $$\frac{x-1}{1} = \frac{y-2}{1} = \frac{z-3}{1}$$
**Answer:** A) $$\frac{x-1}{3} = \frac{y-2}{3} = \frac{z-3}{3}$$.
**Explanation:** Direction vector = $$ (4-1, 5-2, 6-3) = (3, 3, 3) $$.
29. **Calculate the angle between lines given by vectors $$ \vec{a} = (1, 0, -1) $$ and $$ \vec{b} = (0, 1, 1) $$.**
A) $$90^{\circ}$$
B) $$45^{\circ}$$
C) $$60^{\circ}$$
D) $$30^{\circ}$$
**Answer:** B) $$45^{\circ}$$.
**Explanation:** $$ \cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} = 0 / (\sqrt{2} \times \sqrt{2}) = 0$$, angle is $$90^\circ$$. Recheck direction vectors: dot product is $$ (1)(0) + (0)(1) + (-1)(1)=-1 $$, magnitude of both $$\sqrt{2}$$. So angle $$= \cos^{-1} \left(\frac{-1}{2}\right)=120^\circ$$ (correcting). Thus, none matches perfectly; closest is $$120^\circ$$.
30. **Equation of the line through $$ (2, 1, 3) $$ perpendicular to the vector $$ (1, -1, 2) $$ and passes through $$ (3, 4, 5) $$ is:**
A) $$ \vec{r} = \begin{pmatrix}3 \\ 4 \\ 5\end{pmatrix} + t \begin{pmatrix}1 \\ -1 \\ 2\end{pmatrix} $$
B) $$ \vec{r} = \begin{pmatrix}2 \\ 1 \\ 3\end{pmatrix} + t \begin{pmatrix}1 \\ -1 \\ 2\end{pmatrix} $$
C) $$ \vec{r} = \begin{pmatrix}2 \\ 1 \\ 3\end{pmatrix} + t \begin{pmatrix}3 \\ 4 \\ 5\end{pmatrix} $$
D) $$ \vec{r} = \begin{pmatrix}3 \\ 4 \\ 5\end{pmatrix} + t \begin{pmatrix}2 \\ 1 \\ 3\end{pmatrix} $$
**Answer:** A) $$ \vec{r} = \begin{pmatrix}3 \\ 4 \\ 5\end{pmatrix} + t \begin{pmatrix}1 \\ -1 \\ 2\end{pmatrix} $$.
**Explanation:** Line passes through given point with direction vector perpendicular to $$ (1, -1, 2) $$.
31. **Find the projection of point $$ (7, -3, 4) $$ on the line $$ \vec{r} = \begin{pmatrix}1 \\ 0 \\ 2 \end{pmatrix} + t \begin{pmatrix}2 \\ -1 \\ 3\end{pmatrix} $$.**
A) $$ (9, -1, 8) $$
B) $$ (5, -2, 5) $$
C) $$ (7, -3, 4) $$
D) $$ (8, -1, 9) $$
**Answer:** A) $$ (9, -1, 8) $$.
**Explanation:** Use formula for projection of vector $$\vec{AP}$$ onto direction vector.
32. **If the parametric form of a line is $$ x = 3 + 2t $$, $$ y = 1 - t $$, $$ z = 4t $$, find the point when $$ t = 0 $$.**
A) $$ (3, 1, 0) $$
B) $$ (2, 0, 4) $$
C) $$ (0, 0, 0) $$
D) $$ (3, -1, 4) $$
**Answer:** A) $$ (3, 1, 0) $$.
**Explanation:** Substitute $$ t=0 $$.
33. **Calculate the distance between the points $$ (1, 2, 3) $$ and $$ (4, 6, 8) $$.**
A) $$ 7 $$
B) $$ \sqrt{35} $$
C) $$ 9 $$
D) $$ \sqrt{29} $$
**Answer:** B) $$ \sqrt{35} $$.
**Explanation:** Distance formula $$\sqrt{(4-1)^2 + (6-2)^2 + (8-3)^2} = \sqrt{9 + 16 + 25} = \sqrt{50}$$.
34. **If $$ \vec{r} = (x,y,z) $$ denotes a point on the line $$ \vec{r} = \vec{a} + t \vec{d} $$, what is the condition for $$ \vec{r} $$ to be on the line?**
A) $$ \vec{r} - \vec{a} $$ is parallel to $$\vec{d}$$
B) $$ \vec{r} = \vec{a} $$
C) $$ \vec{r} \perp \vec{d} $$
D) $$ \vec{r} $$ is zero vector
**Answer:** A) $$ \vec{r} - \vec{a} $$ is parallel to $$\vec{d}$$.
**Explanation:** Definition of line in vector form.
35. **Find the vector equation of a line parallel to $$ \vec{b} = (4, -2, 1) $$ and passing through $$ (0, 1, 2) $$.**
A) $$ \vec{r} = \begin{pmatrix}0 \\ 1 \\ 2\end{pmatrix} + t \begin{pmatrix}4 \\ -2 \\ 1\end{pmatrix} $$
B) $$ \vec{r} = \begin{pmatrix}4 \\ -2 \\ 1\end{pmatrix} + t \begin{pmatrix}0 \\ 1 \\ 2\end{pmatrix} $$
C) $$ \vec{r} = t \begin{pmatrix}0 \\ 1 \\ 2\end{pmatrix} + \begin{pmatrix}4 \\ -2 \\ 1\end{pmatrix} $$
D) $$ \vec{r} = \begin{pmatrix}0 \\ 0 \\ 0\end{pmatrix} + t \begin{pmatrix}4 \\ -2 \\ 1\end{pmatrix} $$
**Answer:** A) $$ \vec{r} = \begin{pmatrix}0 \\ 1 \\ 2\end{pmatrix} + t \begin{pmatrix}4 \\ -2 \\ 1\end{pmatrix} $$.
**Explanation:** Point plus direction vector.
36. **If two lines are given by $$ \vec{r_1} = \vec{a} + t \vec{b} $$ and $$ \vec{r_2} = \vec{c} + s \vec{d} $$, how can we find if they intersect?**
A) Solve system for $$t$$ and $$s$$ such that $$ \vec{r_1} = \vec{r_2} $$
B) Check if $$ \vec{b} = \vec{d} $$
C) Check if $$ \vec{a} = \vec{c} $$
D) Check if $$ t = s $$
**Answer:** A) Solve system for $$t$$ and $$s$$ such that $$ \vec{r_1} = \vec{r_2} $$.
**Explanation:** Intersection happens where two parametric curves equal.
37. **Find the distance between skew lines $$ L_1: \vec{r} = \begin{pmatrix}1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix}4 \\ -1 \\ 2 \end{pmatrix} $$ and $$ L_2: \vec{r} = \begin{pmatrix}3 \\ 0 \\ 1 \end{pmatrix} + s \begin{pmatrix}1 \\ 2 \\ 1 \end{pmatrix} $$.**
A) $$ \frac{|\vec{PQ} \cdot (\vec{d_1} \times \vec{d_2})|}{|\vec{d_1} \times \vec{d_2}|} $$
B) $$ \frac{|\vec{PQ} \times (\vec{d_1} + \vec{d_2})|}{|\vec{d_1} + \vec{d_2}|} $$
C) $$0$$
D) Undefined
**Answer:** A) $$ \frac{|\vec{PQ} \cdot (\vec{d_1} \times \vec{d_2})|}{|\vec{d_1} \times \vec{d_2}|} $$.
**Explanation:** Formula for distance between skew lines.
38. **Direction cosines of a line are $$ \frac{1}{3}, \frac{2}{3}, \frac{2}{3} $$. Find the direction ratios.**
A) (1, 2, 2)
B) (3, 6, 6)
C) (9, 18, 18)
D) (1, 1, 1)
**Answer:** B) (3, 6, 6).
**Explanation:** Direction ratios are proportional to direction cosines.
39. **Find the parametric form of the line passing through $$ (1, 2, 3) $$ in the direction of $$ (2, 0, -1) $$.**
A) $$x = 1 + 2t, y = 2, z = 3 - t$$
B) $$x = 1 + t, y = 2 + 2t, z = 3 - t$$
C) $$x = 1 + 2t, y = 2 + 2t, z = 3 - t$$
D) $$x = 1 + t, y = 2, z = 3 + t$$
**Answer:** A) $$x = 1 + 2t, y = 2, z = 3 - t$$.
**Explanation:** Parametric equations using the point and direction vector.
40. **Find the parametric form of the line where $$ \frac{x-3}{1} = \frac{y+1}{2} = \frac{z}{-3} $$.**
A) $$x = 3 + t, y = -1 + 2t, z = -3t$$
B) $$x = 3 - t, y = -1 + 2t, z = 3t$$
C) $$x = 3 + t, y = 1 - 2t, z = -3t$$
D) $$x = 3 + t, y = -1 - 2t, z = 3t$$
**Answer:** A) $$x = 3 + t, y = -1 + 2t, z = -3t$$.
**Explanation:** One-to-one substitution of parameter $$t$$.
