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FAQs on Class 12 RS Aggarwal Chapter-23 Scalar, or Dot , Product of Vector Solutions - Free PDF Download
1. How do you calculate the scalar product of two vectors when given in their component form (i.e., using î, ĵ, k)?
To find the scalar or dot product of two vectors, a⃗ = a₁î + a₂ĵ + a₃k̂ and b⃗ = b₁î + b₂ĵ + b₃k̂, you multiply their corresponding components and sum the results. The formula is:
a⃗ ⋅ b⃗ = (a₁ * b₁) + (a₂ * b₂) + (a₃ * b₃)
Since î⋅î = ĵ⋅ĵ = k̂⋅k̂ = 1 and î⋅ĵ = ĵ⋅k̂ = k̂⋅î = 0, all the cross-component products become zero, simplifying the calculation.
2. What is the step-by-step method to find the angle between two vectors using their dot product?
You can find the angle (θ) between two non-zero vectors a⃗ and b⃗ by following these steps:
- Step 1: Calculate the dot product of the two vectors, a⃗ ⋅ b⃗.
- Step 2: Find the magnitude of each vector, |a⃗| and |b⃗|.
- Step 3: Use the dot product formula for the angle: cos θ = (a⃗ ⋅ b⃗) / (|a⃗| |b⃗|).
- Step 4: Solve for θ by taking the inverse cosine: θ = cos⁻¹((a⃗ ⋅ b⃗) / (|a⃗| |b⃗|)).
This method is fundamental for solving many problems in RS Aggarwal Chapter 23.
3. What does it mean if the scalar product of two non-zero vectors is zero?
If the scalar (dot) product of two non-zero vectors is zero (i.e., a⃗ ⋅ b⃗ = 0), it signifies that the two vectors are perpendicular (orthogonal) to each other. This is because a⃗ ⋅ b⃗ = |a⃗| |b⃗| cos θ. For the product to be zero, cos θ must be zero, which only happens when the angle θ is 90°. This is a crucial condition used to prove orthogonality in vector problems.
4. What are the key algebraic properties of the scalar product that are essential for solving problems in this chapter?
The scalar product follows several important algebraic properties:
- Commutative Property: The order of vectors does not matter. a⃗ ⋅ b⃗ = b⃗ ⋅ a⃗.
- Distributive Property: The dot product distributes over vector addition. a⃗ ⋅ (b⃗ + c⃗) = a⃗ ⋅ b⃗ + a⃗ ⋅ c⃗.
- Scalar Multiplication Property: A scalar multiple can be grouped with either vector. (λa⃗) ⋅ b⃗ = a⃗ ⋅ (λb⃗) = λ(a⃗ ⋅ b⃗).
- Dot Product with Itself: The dot product of a vector with itself gives the square of its magnitude. a⃗ ⋅ a⃗ = |a⃗|².
5. How do you find the projection of one vector onto another using the scalar product?
The projection of vector a⃗ onto vector b⃗ is a scalar value that represents the 'shadow' or component of a⃗ in the direction of b⃗. The correct formula to find this is:
Projection of a⃗ on b⃗ = (a⃗ ⋅ b⃗) / |b⃗|
A common mistake is dividing by the magnitude of the wrong vector. Always remember to divide by the magnitude of the vector you are projecting onto.
6. Can the scalar product of two vectors be a negative value, and what does that imply?
Yes, the scalar product can be negative. The sign of the dot product reveals the nature of the angle (θ) between the two vectors:
- If a⃗ ⋅ b⃗ > 0, the angle θ is acute (0° ≤ θ < 90°).
- If a⃗ ⋅ b⃗ = 0, the angle θ is a right angle (θ = 90°).
- If a⃗ ⋅ b⃗ < 0, the angle θ is obtuse (90° < θ ≤ 180°).
A negative result means the vectors are pointing in generally opposite directions.
7. What is the physical significance of the scalar product?
The scalar product has a significant physical interpretation, most notably in the calculation of Work Done. If a constant force vector F⃗ acts on an object causing a displacement d⃗, the work done (W) is the scalar product of the force and displacement vectors: W = F⃗ ⋅ d⃗. It effectively multiplies the component of the force that is in the direction of displacement by the magnitude of the displacement, resulting in a scalar quantity (energy).
