Question 13 of 25intermediate🔧 ApplyShort Answer4 marks

Show that the line through the points (4,7,8),(2,3,4)(4, 7, 8), (2, 3, 4) is parallel to the line through the points (−1,−2,1),(1,2,5)(-1, -2, 1), (1, 2, 5).

Correct Answer

The direction ratios of the line through (4,7,8) and (2,3,4) are (2-4, 3-7, 4-8) = (-2, -4, -4). For the line through (-1,-2,1) and (1,2,5), direction ratios are (1-(-1), 2-(-2), 5-1) = (2, 4, 4). These ratios are proportional because (2,4,4) = -1 × (-2,-4,-4). Simplifying, both sets are proportional to (1,2,2). Since direction ratios are proportional, the lines are parallel. Hence, the given lines are parallel.

Exercise: EXERCISE 11.2 | Q: 3 | (Chapter: Page 13)
For More Understanding

Explanation

This solution uses the property that if the direction ratios of two lines are proportional, then the lines are parallel. This is analogous to Example 5 in the provided context, which demonstrates collinearity by checking proportional direction ratios.

Solution Steps

  1. Step 1: Find direction ratios of first line: (2-4, 3-7, 4-8) = (-2, -4, -4).

  2. Step 2: Find direction ratios of second line: (1-(-1), 2-(-2), 5-1) = (2, 4, 4).

  3. Step 3: Observe that (2,4,4) = -1 × (-2,-4,-4), so the direction ratios are proportional.

  4. Step 4: Conclude that the lines are parallel.

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