Lesson 11 of 11 · 15 min
Chapter review
Watch a class
The whole chapter on YouTube
Must-know facts
18 facts
- 01A vector has magnitude and direction; the zero vector alone has no direction.
- 02Position vector of P(x, y, z): xî + yĵ + zk̂, of length √(x² + y² + z²).
- 03Direction cosines l, m, n satisfy l² + m² + n² = 1; direction ratios need not.
- 04AB + BC = AC, and AB + BC + CA = 0 round a closed triangle.
- 05Vector addition is commutative and associative.
- 06â = a/|a| is the unit vector along a non-zero a.
- 07b is collinear with a exactly when b = λa.
- 08Vector from P₁ to P₂ = position vector of P₂ − position vector of P₁.
- 09Section formula: (mb + na)/(m + n) internally, (mb − na)/(m − n) externally; mid-point (a + b)/2.
- 10a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃, a number.
- 11For non-zero vectors, a·b = 0 exactly when a ⊥ b.
- 12Projection of a on b = (a·b)/|b|.
- 13|a ± b|² = |a|² ± 2a·b + |b|².
- 14a × b = |a||b| sin θ n̂, a vector perpendicular to both (right-hand rule).
- 15b × a = −a × b; a × b = 0 for parallel vectors.
- 16î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ.
- 17Parallelogram area |a × b|; triangle area ½|a × b|.
- 18[a b c] = a·(b × c) = 0 exactly when a, b, c are coplanar (JEE).
Common traps
Where marks are lost
Formulas
10 to know
Magnitude
|xî + yĵ + zk̂| = √(x² + y² + z²)
Also the length of a position vector.
Unit vector
â = a/|a|
a ≠ 0.
Section formula
r = (mb + na)/(m + n); external: (mb − na)/(m − n)
PR : RQ = m : n, P and Q at a and b.
Dot product
a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃
A number; 0 for perpendicular vectors.
Angle
cos θ = (a·b)/(|a||b|)
0 ≤ θ ≤ π.
Projection of a on b
a·b̂ = (a·b)/|b|
Signed shadow length.
Cross product
a × b = |a||b| sin θ n̂
Right-handed; as a determinant with first row î, ĵ, k̂.
Areas
parallelogram |a × b|, triangle ½|a × b|
Diagonals d₁, d₂: ½|d₁ × d₂|.
Box product (JEE)
[a b c] = a·(b × c)
Volume of the parallelepiped; 0 means coplanar.
Vector triple product (JEE)
a × (b × c) = (a·c)b − (a·b)c
Lies in the plane of b and c.
Key terms
8 terms
- Vector
- A quantity with magnitude and direction, drawn as a directed line segment.
- Position vector
- The vector from the origin O to a point P.
- Direction cosines
- cos α, cos β, cos γ of the angles a vector makes with the three axes.
- Direction ratios
- Any three numbers proportional to the direction cosines.
- Unit vector
- A vector of magnitude 1.
- Collinear vectors
- Vectors parallel to one line, whatever their lengths or senses.
- Projection
- The signed length of a vector's shadow on a line: (a·b)/|b| on the line of b.
- Scalar triple product
- [a b c] = a·(b × c), the signed volume of the box on a, b, c.