10 Three Dimensional Geometry

Exercise 01

Question:

If a line has the direction ratios −18,12,−4, then what are its direction cosines?

Answer:

Answer: Step 1: Direction ratios are the components of a vector that are expressed as a fraction of the magnitude of the vector.

Step 2: Direction cosines of a vector are the cosines of the angles between the vector and the three coordinate axes.

Step 3: To find the direction cosines of the line, we need to find the angles between the line and the coordinate axes.

Step 4: The angle between the line and the x-axis is given by tanθx = 12/−18 = −2/3.

Step 5: The angle between the line and the y-axis is given by tanθy = −4/12 = −1/3.

Step 6: The angle between the line and the z-axis is given by tanθz = 12/−4 = −3.

Step 7: The direction cosines of the line are cosθx = −2/3, cosθy = −1/3 and cosθz = −3.

Question:

If a line makes angles 90∘,135∘,45∘ with the x,y and Z−axes respectively, find its direction cosines.

Answer:

Answer:

Step 1: The line makes an angle of 90∘ with the x-axis.

Step 2: Therefore, the direction cosine of the line along the x-axis (lx) is 1.

Step 3: The line makes an angle of 135∘ with the y-axis.

Step 4: Therefore, the direction cosine of the line along the y-axis (ly) is -√2/2.

Step 5: The line makes an angle of 45∘ with the z-axis.

Step 6: Therefore, the direction cosine of the line along the z-axis (lz) is √2/2.

Therefore, the direction cosines of the line are lx = 1, ly = -√2/2 and lz = √2/2.

Question:

Find the direction cosines of a line which makes equal angles with the coordinate axes.

Answer:

Step 1: A line which makes equal angles with the coordinate axes is at an angle of 45° with the x-axis and y-axis.

Step 2: The direction cosines of a line at an angle of 45° with the x-axis and y-axis can be calculated using the formula: cosθx = cos45° = √2/2 cosθy = cos45° = √2/2

Step 3: Therefore, the direction cosines of the line which makes equal angles with the coordinate axes are: cosθx = √2/2 cosθy = √2/2

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JEE NCERT Solutions (Mathematics)

01 Relations and Functions

02 Inverse Trigonometric Functions

03 Matrices

04 Determinants

05 Continuity and Differentiability

06 Application of Derivatives

07 Integrals

08 Application of Integrals

09 Vectors

10 Three Dimensional Geometry

11 Linear Programming

12 Probability

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