Vector Algebra Class 12 Maths Chapter 10 Notes

Key Terms in Vector Algebra

Scalar

A quantity that has only magnitude but no direction. Examples: mass, length, time, temperature.

Vector

A quantity that has both magnitude and direction. It is represented by a directed line segment. Examples: velocity, force, displacement.

Position Vector

A vector that represents the position of a point P with respect to a fixed origin O. It is denoted by OP.

Unit Vector

A vector with a magnitude of 1. A unit vector in the direction of vector a is denoted by â and is calculated as â = a / |a|.

Zero Vector (Null Vector)

A vector with zero magnitude and an arbitrary direction. It is denoted by 0.

Collinear Vectors

Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions. Mathematically, a = λb for some scalar λ.

Equal Vectors

Two vectors a and b are equal if they have the same magnitude and the same direction.

Coinitial Vectors

Two or more vectors having the same initial point.

Must-Remember Formulae & Properties

• {"point":"Magnitude of a Vector: For a vector a = xî + yĵ + zk̂, its magnitude is |a| = √(x² + y² + z²)."}
• {"point":"Unit Vector Formula: The unit vector in the direction of a is â = a / |a|."}
• {"point":"Section Formula (Internal Division): The position vector of a point R that divides the line segment joining points P(a) and Q(b) internally in the ratio m:n is r = (mb + na) / (m + n)."}
• {"point":"Dot (Scalar) Product: a · b = |a||b|cos(θ). Also, a · b = a₁b₁ + a₂b₂ + a₃b₃. If a ⊥ b, then a · b = 0."}
• {"point":"Cross (Vector) Product: a × b = |a||b|sin(θ) n̂, where n̂ is a unit vector perpendicular to both a and b. If a || b, then a × b = 0."}
• {"point":"Projection of Vector a on b: This is a scalar value given by (a · b) / |b|."}
• {"point":"Area of Parallelogram: Area = |a × b|, where a and b are adjacent sides."}
• {"point":"Area of Triangle: Area = ½ |a × b|, where a and b are two sides of the triangle."}
• {"point":"Scalar Triple Product: [a b c] = a · (b × c). It represents the volume of the parallelepiped with adjacent edges a, b, and c."}
• {"point":"Condition for Coplanarity: Three vectors a, b, and c are coplanar if their scalar triple product is zero, i.e., [a b c] = 0."}

Understanding Dot Product and Cross Product

Vector multiplication is not as straightforward as scalar multiplication. We define two primary types of products: the Scalar (Dot) Product and the Vector (Cross) Product. The key difference lies in their output and geometric meaning.

The Dot Product, denoted as a · b, results in a scalar quantity. It's a measure of how much one vector extends in the direction of another. Think of it as a projection. Its formula is a · b = |a||b|cos(θ), where θ is the angle between the vectors. A crucial application is testing for orthogonality: if two non-zero vectors a and b are perpendicular, their dot product is zero (since cos(90°) = 0). This is a heavily tested concept. In component form, if a = a₁î + a₂ĵ + a₃k̂ and b = b₁î + b₂ĵ + b₃k̂, then a · b = a₁b₁ + a₂b₂ + a₃b₃.

The Cross Product, denoted as a × b, results in a new vector that is perpendicular to the plane containing a and b. Its direction is given by the Right-Hand Thumb Rule. The magnitude of the cross product, |a × b| = |a||b|sin(θ), represents the area of the parallelogram formed by vectors a and b. If two non-zero vectors are parallel, their cross product is the zero vector (since sin(0°) = 0). The cross product is calculated using a determinant:

a × b = | î ĵ k̂ |
| a₁ a₂ a₃ |
| b₁ b₂ b₃ |

Remember, the dot product gives a number (scalar), while the cross product gives a direction-dependent entity (vector).

Dot Product vs. Cross Product: A Quick Comparison

Worked Mini-Examples

• {"title":"Example 1: Find the angle between two vectors","bodyMarkdown":"Find the angle θ between vectors a = î + 2ĵ - k̂ and b = î - ĵ + k̂.\n\nSolution:\nWe use the dot product formula: cos(θ) = (a · b) / (|a||b|)\n\n1. Calculate a · b:\n a · b = (1)(1) + (2)(-1) + (-1)(1) = 1 - 2 - 1 = -2\n\n2. Calculate magnitudes:\n |a| = √(1² + 2² + (-1)²) = √(1+4+1) = √6\n |b| = √(1² + (-1)² + 1²) = √(1+1+1) = √3\n\n3. Find cos(θ):\n cos(θ) = -2 / (√6 * √3) = -2 / √18 = -2 / (3√2) = -√2 / 3\n\n4. Angle θ:\n θ = arccos(-√2 / 3)"}
• {"title":"Example 2: Find the area of a parallelogram","bodyMarkdown":"Find the area of the parallelogram whose adjacent sides are given by the vectors a = 3î + ĵ + 4k̂ and b = î - ĵ + k̂.\n\nSolution:\nArea = |a × b|\n\n1. Calculate a × b using the determinant:\n a × b = | î ĵ k̂ |\n | 3 1 4 |\n | 1 -1 1 |\n = î((1)(1) - (4)(-1)) - ĵ((3)(1) - (4)(1)) + k̂((3)(-1) - (1)(1))\n = î(1 + 4) - ĵ(3 - 4) + k̂(-3 - 1)\n = 5î + ĵ - 4k̂\n\n2. Calculate the magnitude:\n |a × b| = √(5² + 1² + (-4)²)\n = √(25 + 1 + 16) = √42\n\nArea of the parallelogram is √42 square units."}

Common Exam Traps

Board Exam Pointers:

• Scalar vs. Vector Result: Be extremely careful. If a question asks for the 'projection of a on b', the answer is a scalar value. If it asks for the 'projection vector', the answer is a vector: ((a · b) / |b|²) b.
• Area Calculation: The area of a parallelogram/triangle is |a × b| (a magnitude), not the vector a × b itself. A common mistake is to leave the answer in î, ĵ, k̂ form. Always find the magnitude.
• Determinant Signs: When calculating the cross product, remember the sign convention for the cofactors: + for î, - for ĵ, and + for k̂. A sign error here will fail the entire question.
• Showing Your Work: Always write down the formula you are using before substituting values (e.g., cos(θ) formula, area formula). This can fetch you partial marks even if your final calculation is incorrect.

Practice Questions with Solutions

• If two vectors a and b are such that |a| = 3, |b| = 2 and a · b = 6, what is the angle between them? cos(θ) = (a · b) / (|a||b|) = 6 / (3*2) = 1. So, θ = 0°. The vectors are parallel and in the same direction.
• What does it mean if the scalar triple product [a b c] = 0? It means the three vectors a, b, and c are coplanar; they lie on the same plane. The volume of the parallelepiped formed by them is zero.
• Find a unit vector in the direction of the vector a = 2î + 3ĵ + k̂. |a| = √(2² + 3² + 1²) = √14. The unit vector â = a / |a| = (1/√14)(2î + 3ĵ + k̂).
• What is the value of î · (ĵ × k̂)? ĵ × k̂ = î. So, î · (ĵ × k̂) = î · î = 1. This is an application of the scalar triple product for orthonormal unit vectors.

Frequently Asked Questions

Frequently Asked Questions