Common Mistakes

Traps in 3D Geometry

6 mistake patterns students fall for. 2 high-frequency traps appear in almost every exam.

Confusing direction cosines with direction ratios

Very CommonFORMULA

Direction cosines (l, m, n) satisfy $l^2 + m^2 + n^2 = 1$ , while direction ratios (a, b, c) are any scalar multiples of DCs and need not be normalized.

Why: Students use the terms DCs and DRs interchangeably, or assume DRs are already normalized.

WRONG: Taking DRs (2, 3, 6) and directly using them as DCs without dividing by

\sqrt{4+9+36} = 7

RIGHT: Always normalize:

l = \frac{a}{\sqrt{a^2+b^2+c^2}}

m = \frac{b}{\sqrt{a^2+b^2+c^2}}

n = \frac{c}{\sqrt{a^2+b^2+c^2}}

. Verify

l^2+m^2+n^2 = 1

See pattern: Find Direction Cosines / Direction Ratios →

Wrong formula for shortest distance between skew lines

Very CommonFORMULA

Students either use the wrong vectors in the scalar triple product or confuse the shortest distance formula with the distance from a point to a line.

Why: The formula involves three vector operations (subtraction, cross product, dot product) and students mix up which vectors go where.

WRONG: Computing

\frac{|(\vec{a_1}-\vec{a_2}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{a_1} \times \vec{a_2}|}

or using direction vectors in the numerator instead of the position vector difference.

RIGHT: Lines

\vec{r} = \vec{a_1}+t\vec{b_1}

and

\vec{r} = \vec{a_2}+s\vec{b_2}

. Distance

= \frac{|(\vec{a_2}-\vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|}

. Numerator uses position vectors; denominator uses direction vectors.

See pattern: Shortest Distance Between Lines →

Wrong sign in point-to-plane distance formula

CommonSIGN ERROR

Students drop the absolute value or substitute the point incorrectly, getting a negative or wrong distance.

Why: Rushing through the substitution and forgetting that distance must be non-negative. Sometimes the constant d is taken with the wrong sign.

WRONG: Computing

\frac{ax_1+by_1+cz_1+d}{\sqrt{a^2+b^2+c^2}}

without absolute value, or writing

d

as positive when the plane equation has it as negative.

RIGHT: Rewrite the plane as

ax+by+cz+d = 0

first. Then

\text{distance} = \frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}

. Always take the absolute value.

See pattern: Distance from Point to Plane →

Using parallel condition instead of perpendicular

CommonCONCEPT

Students mix up when to use the dot product being zero (perpendicular) versus the cross product being zero (parallel).

Why: The conditions for parallel planes (normals are parallel) and perpendicular planes (normals are perpendicular) are easily confused, especially under time pressure.

WRONG: Setting

a_1a_2 + b_1b_2 + c_1c_2 = 0

for parallel planes, when this is actually the condition for perpendicular planes.

RIGHT: Parallel:

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

(proportional normals). Perpendicular:

a_1a_2 + b_1b_2 + c_1c_2 = 0

(orthogonal normals).

See pattern: Angle Between Lines or Planes →

Mixing line-plane angle with plane-plane angle

CommonFORMULA

The angle between a line and a plane uses $\sin\theta$ , while the angle between two planes uses $\cos\theta$ . Students apply the wrong trig function.