A vector is a quantity that has both magnitude and direction. In contrast, a scalar has only magnitude. Vectors are used extensively in physics, engineering, and 3D geometry.
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Types of Vectors
- Zero vector (null vector): Magnitude = 0, denoted as 0 (vector)
- Unit vector: Magnitude = 1, written as a-hat
- Equal vectors: Same magnitude and direction
- Negative vector: Same magnitude, opposite direction
- Collinear (parallel) vectors: Vectors along the same or parallel lines
- Coinitial vectors: Vectors with the same starting point
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Components and Notation
A vector in 3D: a = a1 i-hat + a2 j-hat + a3 k-hat
Magnitude: |a| = √(a12 + a22 + a32)
Unit vector: a-hat = a / |a|
Addition: (a1 i + a2 j + a3 k) + (b1 i + b2 j + b3 k) = (a1+b1) i + (a2+b2) j + (a3+b3) k
Scalar multiplication: k × a = (k a1) i + (k a2) j + (k a3) k
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Products of Vectors
- Dot Product (Scalar Product):
- a · b = |a| |b| cos(theta) = a1 b1 + a2 b2 + a3 b3
- a · b = 0 if a and b are perpendicular
- a · a = |a|2
Cross Product (Vector Product):
a × b is a vector perpendicular to both a and b.
|a × b| = |a| |b| sin(theta)
a × b = determinant with i, j, k in first row; a1, a2, a3 in second; b1, b2, b3 in third.
- a × b = 0 if a and b are parallel
- a × b = -(b × a) (anti-commutative)
Scalar Triple Product:
[a b c] = a · (b × c) — gives the volume of the parallelepiped formed by the three vectors.
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Worked Examples
Find the magnitude of a = 2i + 3j - 6k.
|a| = √(4 + 9 + 36) = √(49) = 7
Find the unit vector in the direction of a = i + 2j - 2k.
|a| = √(1 + 4 + 4) = 3. Unit vector = (1/3)i + (2/3)j - (2/3)k
If a = 3i + 4j and b = i - j + k, find a · b.
a · b = 3(1) + 4(-1) + 0(1) = 3 - 4 + 0 = -1
Find the angle between a = i + j and b = j + k.
cos(theta) = (a · b)/(|a| |b|) = (0 + 1 + 0)/(√(2) × √(2)) = 1/2.
So theta = 60 degrees.
Find a × b where a = 2i + j and b = i + j + k.
a × b = |i j k |
|2 1 0 |
|1 1 1 |
= i(1×1 - 0×1) - j(2×1 - 0×1) + k(2×1 - 1×1)
= i(1) - j(2) + k(1) = i - 2j + k
Check if a = 2i + 6j + 3k and b = 4i + 3j - 2k are perpendicular.
a · b = 8 + 18 - 6 = 20 ≠ 0, so they are NOT perpendicular.
Find the area of a parallelogram with adjacent sides a = i + 2j + 3k and b = 3i - 2j + k.
a × b = i(2-(-6)) - j(1-9) + k(-2-6) = i(8) - j(-8) + k(-8) = 8i + 8j - 8k
Area = |a × b| = √(64 + 64 + 64) = √(192) = 8 √(3) sq units.
Area = 8 √(3) square units
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Common mistakes
- Dot product is a scalar, cross product is a vector: Do not mix up the two products.
- Cross product is not commutative: a × b = -(b × a). Order matters!
- Forgetting to find unit vector when direction cosines are asked: Always divide the vector by its magnitude.
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Summary
Vectors have magnitude and direction. Key operations are addition, scalar multiplication, dot product, and cross product. The dot product gives the cosine of the angle between vectors; the cross product gives a vector perpendicular to both. The scalar triple product gives the volume of a parallelepiped.