DPST1013 Chap.2 Vector Geometry and Projections
Vector Geometry and Projections
Dot products turn angles and orthogonality into scalar calculations, while cross products encode perpendicular directions and area in three dimensions. This chapter uses projection to split a vector into parallel and perpendicular components, then applies that decomposition to distances from points to lines and planes.
What this chapter covers
- 01
Lengths and angles
- 02
Dot products and orthogonality
- 03
Vector projections
- 04
Distance to a line
- 05
Cross products and areas
- 06
Scalar triple products and volumes
Worked example · free
Projection with two independent checks
- 1Compute v dot u = 12 and u dot u = 9.
- 1The projection is (12/9)u = (4/3, 8/3, 8/3).
- 1Subtract to obtain the remainder (8/3, -5/3, 1/3).
- 1The remainder dot u is zero, and the two components add to v.
Key terms
- Dot product
- A scalar measuring alignment through lengths and the cosine of the included angle.
- Orthogonal vectors
- Vectors whose dot product is zero.
- Vector projection
- The component of one vector pointing along another non-zero vector.
- Scalar triple product
- A signed volume calculation formed by dotting one vector with the cross product of two others.
Vector Geometry and Projections FAQ
What checks prove a projection is correct?
Add the projection and perpendicular remainder to recover the original vector. Then take the dot product of the remainder with the target direction and confirm it is zero. Passing both checks strongly audits the calculation.
How do I find the distance from a point to a line?
Subtract a known line point from the external point to form a displacement vector. Remove its projection along the line direction. The norm of the remaining perpendicular component is the shortest distance.
Why does cross-product order matter?
Reversing the two inputs reverses the perpendicular direction, so b cross a equals the negative of a cross b. The magnitude is unchanged, which is why area is unaffected even though orientation changes.
How does the dot product determine an angle?
Rearrange the dot-product identity so the cosine equals the dot product divided by the product of the two norms. Confirm that neither vector is zero and that the computed ratio lies between negative one and one. The inverse cosine returns the angle between zero and pi. Use the sign of the dot product as a reasonableness check: positive means acute, zero means perpendicular and negative means obtuse.
What changes when the projection target is not a unit vector?
The target direction appears in both numerator and denominator: multiply the target vector by the dot product with the source, then divide by the target's squared norm. Omitting that denominator incorrectly scales the component unless the target already has length one. You may instead normalise first, but keep track of whether you want a scalar component or the projected vector itself.
How is a point-to-plane distance obtained?
Take a displacement from any known plane point to the external point and project it onto a normal direction. The absolute dot product with a unit normal is the distance. If the normal is not normalised, divide the absolute dot product by its norm. The absolute value is essential because distance is non-negative even though the normal gives an orientation to the signed component.
When should I use a scalar triple product?
Use it when three vectors span a parallelepiped or when coplanarity is in question. Compute one vector dotted with the cross product of the other two. Its absolute value is the volume, while a zero value means the three directions are coplanar. Retain the sign only when orientation matters. Changing the cyclic order preserves the value, whereas swapping two inputs reverses the sign.
How do I find the area of a triangle from three points?
Form two edge vectors from the same vertex, take their cross product and halve its norm. Using edges from different starting points can introduce a sign or translation mistake, so write the chosen vertex explicitly. The cross-product norm gives the parallelogram area; the factor of one half converts it to the triangle. A zero result indicates that the three points are collinear.
Exam move
Draw the target direction before using a projection formula. Compute the two dot products separately, then test both reconstruction and orthogonality. For cross products, write the input order beside the determinant setup and check perpendicularity against both original vectors.
Convert the resulting vector into the quantity requested: a norm for area, an absolute scalar triple product for volume, or a unit normal when direction alone is needed. Pair every formula with a meaning sentence: the dot product measures aligned components, the cross product encodes an oriented perpendicular, and a norm removes direction.
Practise the same data as an angle, distance, area and decomposition question so that the requested output controls the final step. Keep exact radicals until the end, because premature decimals obscure orthogonality and proportionality. During review, perform the cheapest available check before rereading any algebra.
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