Question 3 of 25beginner🔧 ApplyVery Short2 marks

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

Correct Answer

The direction cosines of the line are −911,611,−211\frac{-9}{11}, \frac{6}{11}, \frac{-2}{11}.

Exercise: EXERCISE 11.1 | Q: 3 | (Page Page 5)
For More Understanding

Explanation

This question tests the relationship between direction ratios and direction cosines. From the textbook, direction ratios are numbers proportional to direction cosines. If a, b, c are direction ratios, then the direction cosines are given by dividing each ratio by the square root of the sum of squares of all ratios. This is demonstrated in Example 2 of the provided context.

Solution Steps

  1. Step 1: Given direction ratios are −18,12,−4-18, 12, -4. Let a=−18,b=12,c=−4a = -18, b = 12, c = -4.

  2. Step 2: Calculate a2+b2+c2=(−18)2+(12)2+(−4)2=324+144+16=484=22\sqrt{a^2 + b^2 + c^2} = \sqrt{(-18)^2 + (12)^2 + (-4)^2} = \sqrt{324 + 144 + 16} = \sqrt{484} = 22.

  3. Step 3: The direction cosines are a22,b22,c22=−1822,1222,−422\frac{a}{22}, \frac{b}{22}, \frac{c}{22} = \frac{-18}{22}, \frac{12}{22}, \frac{-4}{22}.

  4. Step 4: Simplifying, we get −911,611,−211\frac{-9}{11}, \frac{6}{11}, \frac{-2}{11}.

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