Question 1 of 25intermediate🔧 ApplyShort Answer4 marks

If a line makes angles 90∘,135∘,45∘90^\circ, 135^\circ, 45^\circ with the x,yx, y and zz-axes respectively, find its direction cosines.

Correct Answer

Let l,m,nl, m, n be the direction cosines of the line. Let α,β\alpha, \beta, and γ\gamma be the angles made by the line with the x,yx, y, and zz-axes respectively.

Given that the line makes angles α=90∘,β=135∘\alpha = 90^\circ, \beta = 135^\circ, and γ=45∘\gamma = 45^\circ with the axes. By definition, the direction cosines are the cosines of these angles. Therefore, we calculate:

l=cos⁡90∘=0l = \cos 90^\circ = 0 m=cos⁡135∘=−cos⁡45∘=−12m = \cos 135^\circ = -\cos 45^\circ = -\frac{1}{\sqrt{2}} n=cos⁡45∘=12n = \cos 45^\circ = \frac{1}{\sqrt{2}}

Thus, the direction cosines of the line are 0,−12,120, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}. We can verify this using the relation l2+m2+n2=1l^2 + m^2 + n^2 = 1:

02+(−12)2+(12)2=0+12+12=10^2 + \left(-\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = 0 + \frac{1}{2} + \frac{1}{2} = 1
Exercise: EXERCISE 11.1 | Q: 1 | (Page Page 5)
For More Understanding

Explanation

The solution follows the method shown in Example 1 of the provided text, where direction cosines are found by taking the cosine of the given angles. The definition states that direction cosines are the cosines of the angles made by the line with the positive directions of the coordinate axes. The calculated values satisfy the property l2+m2+n2=1l^2 + m^2 + n^2 = 1 mentioned in the text.

Solution Steps

  1. Step 1: Identify the angles made by the line with the axes: α=90∘\alpha = 90^\circ, β=135∘\beta = 135^\circ, γ=45∘\gamma = 45^\circ.

  2. Step 2: Apply the definition of direction cosines: l=cos⁡αl = \cos \alpha, m=cos⁡βm = \cos \beta, n=cos⁡γn = \cos \gamma.

  3. Step 3: Calculate the values: l=cos⁡90∘=0l = \cos 90^\circ = 0, m=cos⁡135∘=−12m = \cos 135^\circ = -\frac{1}{\sqrt{2}}, n=cos⁡45∘=12n = \cos 45^\circ = \frac{1}{\sqrt{2}}.

  4. Step 4: Verify the result using the identity l2+m2+n2=1l^2 + m^2 + n^2 = 1.

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