Three Dimensional Geometry: common misconceptions

If your child says… “The direction cosines themselves add to one.”

Ask together

Does Pythagoras add the coordinate changes themselves, or their squares?

What’s true

Pythagoras makes the squared cosines total one, rather than the cosines themselves.

If your child says… “The starting point gets multiplied by t.”

Ask together

When t is zero, which part must remain to locate the starting point?

What’s true

The parameter multiplies the direction, while the starting point stays fixed.

If your child says… “I add the endpoints to find the direction.”

Ask together

Which operation gives the movement that takes the starting point to the destination?

What’s true

Subtracting the starting coordinates from the destination gives the movement between them.

If your child says… “The dot product alone gives the angle's cosine.”

Ask together

Would doubling the direction numbers turn either line?

What’s true

Dividing by both direction lengths removes their scale, and the absolute value selects the smaller angle.

If your child says… “If two lines aren't parallel, they must meet.”

Ask together

How could extending either right-hand line take it out of its own plane?

What’s true

In space, nonparallel lines can be skew, with no shared point or shared plane.

If your child says… “Any two chosen points give the distance between the lines.”

Ask together

Does AB cross both lines at right angles, as MN does?

What’s true

For skew or parallel lines, the shortest connector must meet both lines at right angles.

If your child says… “I square all three coordinates to find the distance from an axis.”

Ask together

Does the y-coordinate change between P and its foot?

What’s true

The nearest axis point keeps the along-axis coordinate, so only the other coordinates contribute.

If your child says… “The nearest point on the mirror line is the image.”

Ask together

If the image stopped at M, would M be halfway between the two points?

What’s true

The foot lies halfway between the original point and its image.