FAQs on Equation of a Line in Space - Three Dimensional Geometry, Class 12, Math

Question 1

1. What is the equation of a line in three-dimensional space?

Accepted Answer

Ans. The equation of a line in three-dimensional space can be represented in vector form as r = a + tb, where r is the position vector of a point on the line, a is the position vector of a known point on the line, t is a scalar parameter, and b is the direction vector of the line.

Question 2

2. How can I find the direction vector of a line in three-dimensional space?

Accepted Answer

Ans. To find the direction vector of a line in three-dimensional space, you can subtract the position vectors of two known points on the line. For example, if two points A and B are on the line, the direction vector can be found as b = B - A.

Question 3

3. What is the significance of the scalar parameter t in the equation of a line in three-dimensional space?

Accepted Answer

Ans. The scalar parameter t in the equation of a line in three-dimensional space represents the position of a point on the line relative to the known point a. By varying the value of t, you can obtain different points on the line. For example, when t = 0, the point on the line is at a, and when t = 1, the point is at a + b.

Question 4

4. Can the equation of a line in three-dimensional space be written in Cartesian form?

Accepted Answer

Ans. Yes, the equation of a line in three-dimensional space can also be written in Cartesian form as a set of three equations. The x, y, and z coordinates of a point on the line can be expressed in terms of the parameters t and the components of the direction vector. These equations represent the intersection of the line with the three coordinate planes.

Question 5

5. How can I determine if two lines in three-dimensional space are parallel or intersecting?

Accepted Answer

Ans. Two lines in three-dimensional space are parallel if their direction vectors are scalar multiples of each other. If the direction vectors are not scalar multiples, the lines are either intersecting or skew (not lying in the same plane). To determine if they intersect, you can set up the equations of the lines and solve for the parameters. If the parameters have consistent solutions, the lines intersect at a point.

Equation of a Line in Space - Three Dimensional Geometry, Class 12, Math

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