Question

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

Hint:

Form direction vectors from each pair of points and check their dot product is zero.

Answer 1

Step 1: Using direction cosines instead:
Direction ratios of line 1 are \((2,5,-4)\) with \(|\vec d_1|=\sqrt{4+25+16}=\sqrt{45}\); of line 2, \((3,2,4)\) with \(|\vec d_2|=\sqrt{9+4+16}=\sqrt{29}\).

Step 2: Computing cos(angle) between them:
\(\cos\theta=\dfrac{\vec d_1\cdot\vec d_2}{|\vec d_1||\vec d_2|}=\dfrac{6+10-16}{\sqrt{45}\sqrt{29}}=\dfrac{0}{\sqrt{1305}}=0\).

Step 3: Interpreting the angle:
\(\cos\theta=0\Rightarrow\theta=90^\circ\).

Final Answer:
The angle between the lines is \(90^\circ\), so \(\boxed{\text{they are perpendicular}}\), matching the direct dot-product method.