Question 16 of 25intermediate🔧 ApplyShort Answer4 marks

Show that the lines x−57=y+2−5=z1\frac{x-5}{7} = \frac{y+2}{-5} = \frac{z}{1} and x1=y2=z3\frac{x}{1} = \frac{y}{2} = \frac{z}{3} are perpendicular to each other.

Correct Answer

The given lines are:

Line 1: x−57=y+2−5=z1\frac{x-5}{7} = \frac{y+2}{-5} = \frac{z}{1}

Line 2: x1=y2=z3\frac{x}{1} = \frac{y}{2} = \frac{z}{3}

For a line in the form x−x0a=y−y0b=z−z0c\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}, the direction ratios are (a,b,c)(a, b, c).

Direction ratios of Line 1: (7,−5,1)(7, -5, 1)

Direction ratios of Line 2: (1,2,3)(1, 2, 3)

Two lines are perpendicular if the dot product of their direction ratios equals zero.

Dot product: =7(1)+(−5)(2)+1(3)=7−10+3=0= 7(1) + (-5)(2) + 1(3) = 7 - 10 + 3 = 0

Since the dot product of the direction ratios is zero, the two lines are perpendicular to each other.

Exercise: EXERCISE 11.2 | Q: 11 | (Page Page 14)
For More Understanding

Explanation

This question tests the understanding of direction ratios and the condition for perpendicularity of lines. From the textbook context, lines in symmetric form have direction ratios given by the denominators. The dot product condition (sum of products of corresponding direction ratios equals zero) determines perpendicularity. This mirrors similar problems in Exercise 11.2 where students find angles between lines or values of parameters for perpendicular lines.

Solution Steps

  1. Step 1: Identify direction ratios of Line 1 as (7, -5, 1)

  2. Step 2: Identify direction ratios of Line 2 as (1, 2, 3)

  3. Step 3: Calculate dot product: 7(1) + (-5)(2) + 1(3)

  4. Step 4: Simplify: 7 - 10 + 3 = 0

  5. Step 5: Conclude lines are perpendicular since dot product equals zero

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