Vectors
What this domain tests
Vector algebra in the plane and in space: components, magnitude, collinearity, position vectors, the scalar (dot) product, angles, perpendicularity, projections and simple geometric configurations such as midpoints and points dividing a segment in a ratio.
Core rules
- A vector has magnitude and direction. Equal vectors share both; a position vector OA joins the origin to A(a₁, a₂).
- Components: a = (a₁, a₂), |a| = √(a₁² + a₂²); the unit vector in the direction of a is a/|a|.
- Collinearity: b = λa for some real λ; in components a₁b₂ − a₂b₁ = 0. Parallel vectors have proportional components, not necessarily equal ones.
- Dot product: a·b = a₁b₁ + a₂b₂ = |a||b|cos θ. Hence a ⊥ b ⟺ a·b = 0, cos θ = (a·b)/(|a||b|), and a·a = |a|².
- Midpoint of AB: position vector (OA + OB)/2. Point P dividing AB with AP : PB = m : n has OP = (n·OA + m·OB)/(m + n).
- Projection of a onto b: scalar a·b/|b|, vector (a·b/|b|²)b.
Worked example
Given a = (1, 2) and b = (3, −1), find λ so that a + λb is perpendicular to a − b.
Compute a + λb = (1 + 3λ, 2 − λ) and a − b = (1 − 3, 2 + 1) = (−2, 3).
The perpendicularity condition is a zero dot product:
(−2)(1 + 3λ) + 3(2 − λ) = −2 − 6λ + 6 − 3λ = 4 − 9λ.
Setting 4 − 9λ = 0 gives λ = 4/9. As a check, a + (4/9)b = (7/3, 14/9), and its dot product with (−2, 3) is −14/3 + 14/3 = 0, confirming perpendicularity.
Traps that cost marks
- Writing |a + b| = |a| + |b|; magnitudes add only for parallel same-direction vectors.
- Using the dot product where a scalar multiple is needed, or treating a·b as a vector.
- Forgetting that a negative dot product means an obtuse angle, not an error.
- Applying the midpoint formula when the question divides the segment in a ratio - the coefficients cross over (n with OA, m with OB).
- Checking perpendicularity with components swapped rather than multiplied and added.