Multiplication of Vector
by a Scalar
Scaling vectors, changing direction, unit vector derivation — all applications of multiplying a vector by a scalar.
When a vector a is multiplied by a real number (scalar) λ, the result λa is a new vector whose:
= |λ| × |a|
(absolute value of scalar × original magnitude)
Same as a if λ > 0
Opposite to a if λ < 0
Zero vector if λ = 0
λa = λa₁î + λa₂ĵ + λa₃k̂
Magnitude: |λa| = |λ| · |a|
a = 2î − 3ĵ + k̂, λ = −4
λa = −4(2î − 3ĵ + k̂) = −8î + 12ĵ − 4k̂
|a| = √(4+9+1) = √14
|λa| = |−4|·√14 = 4√14
1. λ(a + b) = λa + λb (distributive over vector sum)
2. (λ + μ)a = λa + μa (distributive over scalar sum)
3. λ(μa) = (λμ)a (associativity with scalar product)
4. 1 · a = a (multiplicative identity)
5. (−1) · a = −a (negative vector)
6. 0 · a = 0 (zero scalar gives zero vector)
Boards ask you to prove the distributive law geometrically. Show that both sides give the same resultant using the triangle or parallelogram law.
The unit vector â is simply a multiplied by the scalar 1/|a|:
Conversely: a = |a| · â
(any vector = its magnitude × its unit vector)
a = 3î + 4ĵ − 0k̂
|a| = 5
â = (1/5)(3î + 4ĵ) = (3/5)î + (4/5)ĵ
Verify: a = 5 · [(3/5)î + (4/5)ĵ] = 3î + 4ĵ ✓
A linear combination of vectors a, b, c is an expression of the form λa + μb + νc where λ, μ, ν are scalars.
r = λa + μb + νc — any vector in 3D space
Standard basis: r = xî + yĵ + zk̂ (linear combo of î, ĵ, k̂)
Linear combinations are used to express unknown vectors in terms of known ones. If three non-coplanar vectors span 3D space, any vector can be uniquely written as their linear combination. This underpins the entire coordinate system.
Since scalar multiplication scales a vector along its direction, collinear vectors are precisely scalar multiples:
λ > 0: same direction | λ < 0: opposite direction
Find λ if vectors a = 2î + λĵ − k̂ and b = 4î − 2ĵ + 2k̂ are collinear.
For collinearity: a₁/b₁ = a₂/b₂ = a₃/b₃
2/4 = λ/(−2) = (−1)/2
1/2 = λ/(−2) → λ = −1
Three vectors a, b, c are coplanar if one can be expressed as a linear combination of the other two:
Equivalent: scalar triple product = 0
[a b c] = a · (b × c) = 0
Coplanarity is tested using the scalar triple product (box product). If [ a b c ] = 0, the vectors are coplanar. This is a major formula in JEE 3D geometry and vector problems.