### 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