Show that the line through the points $(1, -1, 2), (3, 4, -2)$ is per…

Question 12 of 25intermediate🔧 ApplyShort Answer4 marks

Show that the line through the points $(1, -1, 2), (3, 4, -2)$ is perpendicular to the line through the points $(0, 3, 2)$ and $(3, 5, 6)$ .

Correct Answer

Let the line passing through the points $(1, -1, 2)$ and $(3, 4, -2)$ have direction ratios proportional to $(3-1, 4-(-1), -2-2)$ , which simplifies to $(2, 5, -4)$ . Let the line passing through the points $(0, 3, 2)$ and $(3, 5, 6)$ have direction ratios proportional to $(3-0, 5-3, 6-2)$ , which simplifies to $(3, 2, 4)$ .

To determine if the lines are perpendicular, we check if the sum of the products of their corresponding direction ratios is zero. Calculating this sum: $(2)(3) + (5)(2) + (-4)(4) = 6 + 10 - 16 = 0$ . Since the result is zero, the two lines are perpendicular to each other.

Exercise: EXERCISE 11.2 | Q: 2 | (Chapter: Page 13)

For More Understanding

Explanation

The solution follows the method demonstrated in Example 3 of the provided text, which establishes that the direction ratios of a line passing through two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are proportional to $(x_2-x_1), (y_2-y_1), (z_2-z_1)$ . Once the direction ratios for both lines are found, the condition for perpendicularity (dot product equals zero) is applied to complete the proof as requested in the textbook question.

Solution Steps

Step 1: Find direction ratios of the first line connecting $(1, -1, 2)$ and $(3, 4, -2)$ . The ratios are $(3-1, 4-(-1), -2-2) = (2, 5, -4)$ .
Step 2: Find direction ratios of the second line connecting $(0, 3, 2)$ and $(3, 5, 6)$ . The ratios are $(3-0, 5-3, 6-2) = (3, 2, 4)$ .
Step 3: Calculate the sum of the products of corresponding direction ratios: $2(3) + 5(2) + (-4)(4)$ .
Step 4: Simplify the expression: $6 + 10 - 16 = 0$ . Since the sum is zero, the lines are perpendicular.

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