Vector algebra

The essentials

Vectors are added by the triangle law: place the tail of b at the head of a; the vector from the tail of a to the head of b is a + b. Equivalently, by the parallelogram law, a + b is the diagonal of the parallelogram with sides a and b. Addition is commutative and associative. In components, (a₁i + a₂j + a₃k) + (b₁i + b₂j + b₃k) = (a₁ + b₁)i + (a₂ + b₂)j + (a₃ + b₃)k.

Scalar multiplication: λa has magnitude |λ||a|; it points the same way as a if λ > 0 and the opposite way if λ < 0. Two non-zero vectors a and b are collinear exactly when b = λa for some scalar λ, that is, when their components are proportional.

Key points

• The vector joining P(x₁, y₁, z₁) to Q(x₂, y₂, z₂) is (x₂ − x₁)i + (y₂ − y₁)j + (z₂ − z₁)k.
• Section formula (internal, ratio m : n): the point R dividing PQ has position vector (mb + na)/(m + n), where a and b are the position vectors of P and Q.
• Section formula (external, ratio m : n): (mb − na)/(m − n).
• Midpoint of PQ: (a + b)/2.
• Example: λi + 6j − 2k is parallel to 2i + 3j − k when λ/2 = 6/3 = −2/−1, so λ = 4.

Exam tips

• In the section formula, the ratio number m goes with the FAR end point b. Check with the midpoint case m = n.
• For collinearity of three points A, B, C, show that AB and AC are parallel.

Common mistakes

• Swapping m and n in the section formula.
• Using the internal formula when the question says "externally".