Q.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 $(3, 5, 6)$.

Concept: Perpendicular Vectors Condition — two lines are perpendicular if the dot product of their direction vectors is zero.

Step 1: Direction vector of first line

$\vec{d_1} = (3-1,\; 4-(-1),\; -2-2) = (2, 5, -4)$

Step 2: Direction vector of second line

$\vec{d_2} = (3-0,\; 5-3,\; 6-2) = (3, 2, 4)$

Step 3: Dot product

$\vec{d_1} \cdot \vec{d_2} = 2(3) + 5(2) + (-4)(4) = 6 + 10 - 16 = 0$

Since the dot product is zero, the direction vectors are perpendicular, hence the lines are perpendicular.

Concept and Intuition

The condition for two lines to be perpendicular in 3D space is not about slopes (as in 2D) but about their direction vectors. A line's direction is captured by the vector from one point to another along it. If we take the direction vectors $\vec{d_1}$ and $\vec{d_2}$ of the two lines, the lines are perpendicular exactly when these vectors are orthogonal — meaning their dot product is zero:

This works because the dot product measures how much two vectors point in the same direction. When it's zero, they point at right angles. The actual positions of the points don't matter — only the direction matters for perpendicularity.

Step-by-step Solution

1. Find the direction vector of the first line.

The first line passes through $A(1, -1, 2)$ and $B(3, 4, -2)$. The direction vector is simply $B - A$:

2. Find the direction vector of the second line.

The second line passes through $C(0, 3, 2)$ and $D(3, 5, 6)$. Its direction vector is $D - C$:

3. Compute the dot product of the two direction vectors.

4. Interpret the result.

Since the dot product is zero, the direction vectors are perpendicular. Therefore, the lines themselves are perpendicular.

Method: Perpendicularity of lines given by two points each

When each line is specified by two points, first convert to a direction vector, then apply the right-angle test. Positions are irrelevant to perpendicularity — only directions matter.

Steps

Step 1: Build each direction vector as the difference of that line's two points:

Step 2: Apply the dot-product test. The lines are perpendicular iff

Step 3: Compute carefully, signs included. A single sign slip (e.g. $(-4)(4)=-16$, not $+16$) can turn a true zero into a false non-zero. Add the three products and compare with $0$.

Step 4: Conclude. A zero sum proves perpendicularity — no need to check whether the lines actually meet, since even non-intersecting (skew) lines can be perpendicular in direction.

The technique is identical for lines given in symmetric or vector form; only Step 1 (how you read the direction) changes.

Common Mistakes

Mistake 1: Using the given points directly instead of the direction vectors.

Why it's wrong: perpendicularity depends on direction, so you must first subtract to get $\vec d_1=(2,5,-4)$ and $\vec d_2=(3,2,4)$. Correct approach: form each direction as the difference of that line's two points, then dot.

Mistake 2: A sign error in the dot product.

Why it's wrong: $(-4)(4)=-16$, not $+16$; getting this wrong turns the true $0$ into a false non-zero. Correct approach: compute $6+10-16=0$ term by term.

Mistake 3: Trying to check whether the lines intersect.

Why it's wrong: two lines can be perpendicular without meeting (skew). Correct approach: a zero dot product alone settles perpendicularity — intersection is irrelevant.