Vector Algebra: Key Concepts and Exam Applications: quick revision, 15 bites

Dot vs Cross Product: Two Ways to Multiply Vectors

One gives a scalar, the other a vector.

Remember: **dot** is for **scalar** results (angle, projection), **cross** is for **vector** results (area, perpendicular).

Takeaway. Dot product answers 'how much in the same direction?' while cross product answers 'how much perpendicular?'.

Angle Between Two Vectors

The cosine formula you must memorize.

Takeaway. Rearrange to find angle: θ = cos⁻¹( (a·b)/(|a||b|) ).

Projection of a Vector

Projection is just dot product divided by magnitude.

Takeaway. Projection is a scalar length; the projection vector is that length times the unit vector of b.

Area of Parallelogram and Triangle

Cross product magnitude gives area directly.

Takeaway. For triangle with vertices A, B, C, area = ½ |AB × AC|.

Collinearity and Coplanarity via Vectors

Cross product zero? They're parallel.

Takeaway. Zero cross product means parallel; zero scalar triple product means coplanar.

Don't Confuse Projection with Projection Vector

Scalar vs vector: know the difference.

Takeaway. Projection is a number; projection vector is a vector along b.

Cross Product is Not Commutative

Order matters for cross product.

Takeaway. Cross product is anti-commutative; reversing order reverses direction.

Find Area of Triangle with Given Vertices

Use vectors to compute area easily.

Takeaway. Area = ½ |AB × AC|. Compute cross product, then magnitude, then halve.

Unit Vector Perpendicular to Two Vectors

Cross product gives perpendicular direction.

Takeaway. Unit vector perpendicular to both a and b is (a × b) / |a × b|.

Quick Check: |a·b| = |a×b| Implies θ = 45°

A common exam shortcut.

If |a·b| = |a×b|, then |a||b|cosθ = |a||b|sinθ → tanθ = 1 → θ = 45° (or π/4).

Takeaway. Equality of dot and cross magnitudes means angle is 45°.

Remember: Dot for Angle, Cross for Area

Two products, two purposes.

**Dot** → **D**irection (angle) · **Cross** → **C**over (area).

Takeaway. Use dot for angle/projection, cross for area/perpendicular.

Scalar Triple Product Tests Coplanarity

Zero triple product? Coplanar.

Three vectors a, b, c are coplanar if their scalar triple product [a b c] = a·(b×c) = 0.

Takeaway. Coplanarity ⇔ scalar triple product = 0.

Direction Cosines from Vector Components

Cosines of angles with axes.

For vector a = x i + y j + z k, direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1.

Takeaway. Direction cosines are components divided by magnitude.

Don't Forget to Normalize for Unit Vector

Unit vector needs magnitude 1.

Takeaway. Unit vector = vector divided by its length.

Use Determinant for Cross Product

Set up a 3x3 determinant.

For a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, a × b = determinant of [[i, j, k], [a₁, a₂, a₃], [b₁, b₂, b₃]].

Takeaway. Determinant method avoids sign errors in cross product.