Question

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

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Hint: First see the equation of line passing through the points or see the direction ratio of the lines of two lines through the points given above and then use the condition of two perpendicular lines. Use the concept of dot and cross product.

First let's name the points given, A= (1, -1, 2) and B = (3, 4, -2)
C = (0, 3, 2) and D = (3, 5, 6)
Two lines with direction ratio ${a_1},{b_1},{c_1}$ and ${a_2},{b_2},{c_2}$are said to be perpendicular to each other if
${a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0$………(1)
Direction ratio of the line passing through the points is given by $({x_2} - {x_1}),({y_{_2}} - {y_1}),({z_2} - {z_1})$
∴ Direction ratio of the line passing through the points A and B is
$(3 - 1),(4 - ( - 1)),( - 2 - 2)$
$\Rightarrow$2, 5, -4
∴${a_1}$ = 2, ${b_1}$ = 5, ${c_1}$=-4
Direction ratio of the line passing through the points C and D is
$(3 - 0),(5 - 3),(6 - 2)$
$\Rightarrow$3, 2, 4
$\Rightarrow$${a_2}$ = 3, ${b_2}$ = 2, ${c_2}$ = 4
Now using equation (1) we get
= (2×3) + (5×2) + (-4×-4)
=6 + 10 + (-16)
=16 – 16
= 0.
Since the equation (1) is satisfied which shows that the two lines passing through the above points are perpendicular to each other

Note: This question can also be solved using the dot product, for which you first have to make these points into vector equations of two lines and then the dot product of the two lines which should be equal to 0 to show that they are perpendicular.