Lesson 11 of 11 · 15 min

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Must-know facts

18 facts

1A vector has magnitude and direction; the zero vector alone has no direction.
2Position vector of P(x, y, z): xî + yĵ + zk̂, of length √(x² + y² + z²).
3Direction cosines l, m, n satisfy l² + m² + n² = 1; direction ratios need not.
4AB + BC = AC, and AB + BC + CA = 0 round a closed triangle.
5Vector addition is commutative and associative.
6â = a/|a| is the unit vector along a non-zero a.
7b is collinear with a exactly when b = λa.
8Vector from P₁ to P₂ = position vector of P₂ − position vector of P₁.
9Section 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

Writing a·b as a vector or a × b as a number.

The dot product is a scalar; the cross product is a vector (its length is a number).

Taking a × b = b × a.

Swapping the order flips the direction: b × a = −(a × b).

Forgetting the ½ for the area of a triangle.

Parallelogram: |a × b|. Triangle: ½|a × b|.

Using (mb + na)/(m + n) for an external division.

Externally the signs change: (mb − na)/(m − n).

Swapping m and n in the section formula.

For PR : RQ = m : n, m goes with Q's vector b and n with P's vector a.

Saying direction ratios satisfy a² + b² + c² = 1.

Only direction cosines do; divide the ratios by √(a² + b² + c²) to get them.

Finding the vector PQ as P − Q.

It is head minus tail: position vector of Q minus that of P.

Giving only one unit vector perpendicular to two vectors.

There are two, ±(a × b)/|a × b|.

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.