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: Understanding the Concept:
Two lines are perpendicular exactly when their direction vectors have a zero dot product.

Step 2: Finding the first direction vector:
For line 1: \(\vec d_1=(3-1,\,4-(-1),\,-2-2)=(2,5,-4)\).

Step 3: Finding the second direction vector:
For line 2: \(\vec d_2=(3-0,\,5-3,\,6-2)=(3,2,4)\).

Step 4: Computing the dot product:
\(\vec d_1\cdot\vec d_2=(2)(3)+(5)(2)+(-4)(4)=6+10-16=0\).

Final Answer:
Since \(\vec d_1\cdot\vec d_2=0\), \(\boxed{\text{the two lines are perpendicular}}\).