Vectors

Question 1: Basic Vector Operations

Problem Choose the set of correct options using Figure M2W1AQ1. (The figure shows points A(1, 2) and B(2, 3) which can be represented by vectors A = (1, 2) and B = (2, 3)).

Options

• 2A is the vector (2, 4).
• 3B is the vector (6, 9).
• A + B is the vector (3, 5). (Assuming a typo in the original option)
• A - B is the vector (-1, -1).

Correct Options

• 2A is the vector (2, 4).
• 3B is the vector (6, 9).
• A + B is the vector (3, 5).
• A - B is the vector (-1, -1).

Concepts Explained 💡

Vector operations are performed coordinate-wise.

• Scalar Multiplication: To multiply a vector by a scalar (a number), you multiply each component of the vector by that scalar.
  • k * (x, y) = (k*x, k*y)
• Vector Addition/Subtraction: To add or subtract vectors, you add or subtract their corresponding components.
  • (x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂)
  • (x₁, y₁) - (x₂, y₂) = (x₁ - x₂, y₁ - y₂)

Step-by-Step Solution

• 2A: 2 * (1, 2) = (2 * 1, 2 * 2) = (2, 4)
• 3B: 3 * (2, 3) = (3 * 2, 3 * 3) = (6, 9)
• A + B: (1, 2) + (2, 3) = (1 + 2, 2 + 3) = (3, 5)
• A - B: (1, 2) - (2, 3) = (1 - 2, 2 - 3) = (-1, -1)

All the statements are correct.

Question 2: Linear Combination of Vectors

Problem Let V₁ = (1, 1), V₂ = (1, 0), and V₃ = (0, 1) be three vectors. Find out the correct set of options.

Options

• (2, 3) = 2V₁ + 0V₂ + V₃
• (2, 3) = 0V₁ + 2V₂ + 3V₃
• (2, 3) = 2V₁ + V₂ + 0V₃
• (2, 3) = 0V₁ + 3V₂ + 2V₃

Correct Options

• (2, 3) = 2V₁ + 0V₂ + V₃
• (2, 3) = 0V₁ + 2V₂ + 3V₃

Concepts Explained 💡

A linear combination of vectors is an expression constructed from a set of vectors by multiplying each vector by a scalar and adding the results. To check if an equation is true, simply calculate the right-hand side and see if it equals the left-hand side.

Step-by-Step Solution

We evaluate the right-hand side for each option:

• Option 1: 2(1, 1) + 0(1, 0) + 1(0, 1) = (2, 2) + (0, 0) + (0, 1) = (2, 3). This is correct.
• Option 2: 0(1, 1) + 2(1, 0) + 3(0, 1) = (0, 0) + (2, 0) + (0, 3) = (2, 3). This is correct.
• Option 3: 2(1, 1) + 1(1, 0) + 0(0, 1) = (2, 2) + (1, 0) + (0, 0) = (3, 2). This is incorrect.
• Option 4: 0(1, 1) + 3(1, 0) + 2(0, 1) = (0, 0) + (3, 0) + (0, 2) = (3, 2). This is incorrect.

Question 3, 4, 5: Vectors in Data Representation

This set of questions refers to the following table of marks:

Question 3: Vector Representation

Problem Choose the following set of correct options.

• Marks obtained by Romy in Quiz 1, Quiz 2 and End sem represent a row vector.
• Quiz 2 marks of Karthika, Romy and Farzana represent a column vector.
• Number of components in column vector representing Quiz 2 marks are 9.
• Number of components in row vector representing Romy’s marks are 3.

Correct Options

• Marks obtained by Romy in Quiz 1, Quiz 2 and End sem represent a row vector.
• Quiz 2 marks of Karthika, Romy and Farzana represent a column vector.
• Number of components in row vector representing Romy’s marks are 3.

Explanation

• Romy’s marks across the exams can be written as [33, 41, 45], which is a row vector with 3 components.
• The Quiz 2 marks for all students can be written as [50, 41, 21]ᵀ, which is a column vector with 3 components.
• The statement that the Quiz 2 vector has 9 components is incorrect.

Question 4: Scalar Multiplication

Problem In order to improve her marks, Farzana undertook project work and succeeded in increasing her marks. Her marks became doubled for each exam. Choose the correct options.

• To obtain the marks obtained by Farzana after completion of the project, scalar multiplication has to be done by 2 to the row vector representing Farzana’s marks.
• After completion of the project the row vector representing Farzana’s marks is (76, 42, 70).

Correct Options

• To obtain the marks obtained by Farzana after completion of the project, scalar multiplication has to be done by 2 to the row vector representing Farzana’s marks.
• After completion of the project the row vector representing Farzana’s marks is (76, 42, 70).

Explanation

• Farzana’s initial marks vector is F = [38, 21, 35].
• Doubling her marks means performing a scalar multiplication by 2.
• The new marks vector is 2 * F = 2 * [38, 21, 35] = [76, 42, 70].

Question 5: Vector Addition

Problem Following Farzana’s improved marks (doubled for each exam), all students were given bonus marks in Quiz 2, given by the column vector [10, 12, 15]ᵀ. What will be the column vector representing the final marks obtained in Quiz 2 by Karthika, Romy and Farzana?

Correct Option

• [60, 53, 57]ᵀ

Explanation

1. Initial Quiz 2 Marks: The column vector for original Quiz 2 marks is Q₂_initial = [50, 41, 21]ᵀ.
2. Farzana’s Improvement: Farzana’s Quiz 2 mark is doubled: 21 * 2 = 42. The marks vector before the bonus is Q₂_improved = [50, 41, 42]ᵀ.
3. Add Bonus Marks: Add the bonus vector to the improved marks vector. Final Marks = Q₂_improved + Bonus = [50, 41, 42]ᵀ + [10, 12, 15]ᵀ = [50+10, 41+12, 42+15]ᵀ = [60, 53, 57]ᵀ.

Question 6: Vector Identities

Problem Let A and B be two vectors. Which of the following statements is (are) true?

• 3A + 5B = 3(A + B) + [(A + B) - (A - B)]
• 3A + 5B = 5(A + B) - [(A + B) - (A - B)]
• 3A + 5B = 3(A + B) + [(A + B) + (A - B)]
• 3A + 5B = 5(A + B) - [(A + B) + (A - B)]

Correct Options

• 3A + 5B = 3(A + B) + [(A + B) - (A - B)]
• 3A + 5B = 5(A + B) - [(A + B) + (A - B)]

Concepts Explained 💡

To verify vector identities, simplify the right-hand side (RHS) using basic vector algebra and check if it equals the left-hand side (LHS). Two helpful simplifications are:

• (A + B) + (A - B) = 2A
• (A + B) - (A - B) = 2B

Step-by-Step Solution

• Option 1: RHS = 3(A + B) + [2B] = 3A + 3B + 2B = 3A + 5B. This matches the LHS. Correct.
• Option 2: RHS = 5(A + B) - [2B] = 5A + 5B - 2B = 5A + 3B. This does not match. Incorrect.
• Option 3: RHS = 3(A + B) + [2A] = 3A + 3B + 2A = 5A + 3B. This does not match. Incorrect.
• Option 4: RHS = 5(A + B) - [2A] = 5A + 5B - 2A = 3A + 5B. This matches the LHS. Correct.

Question 7: Standard Basis Vectors

Problem Let V₁ = (1, 0, 0), V₂ = (0, 1, 0) and V₃ = (0, 0, 1) be three vectors and a, b, c be three real numbers (scalars). Which of the following is (are) true?

• (a, b, c) = aV₁ + bV₂ + cV₃
• (a, b, c) = abV₁ + bcV₂ + caV₃
• (a, 0, c) = aV₁ + cV₂ + 0V₃
• (a, 0, c) = aV₁ + 0V₂ + cV₃

Correct Options

• (a, b, c) = aV₁ + bV₂ + cV₃
• (a, 0, c) = aV₁ + 0V₂ + cV₃

Concepts Explained 💡

The vectors V₁, V₂, and V₃ are the standard basis vectors in 3D space, often denoted as î, ĵ, and k̂. Any vector (x, y, z) can be uniquely expressed as a linear combination xV₁ + yV₂ + zV₃.

Step-by-Step Solution

• Option 1: aV₁ + bV₂ + cV₃ = a(1,0,0) + b(0,1,0) + c(0,0,1) = (a,0,0) + (0,b,0) + (0,0,c) = (a,b,c). Correct.
• Option 2: abV₁ + bcV₂ + caV₃ = (ab, bc, ca). This is not equal to (a,b,c). Incorrect.
• Option 3: aV₁ + cV₂ + 0V₃ = a(1,0,0) + c(0,1,0) = (a, c, 0). This is not equal to (a,0,c). Incorrect.
• Option 4: aV₁ + 0V₂ + cV₃ = a(1,0,0) + 0 + c(0,0,1) = (a, 0, c). Correct.

Question 8: Geometric Interpretation of Vectors

Problem Consider vectors A(-1, 2) and B(2, -2). Choose the set of correct options based on the figure.

Correct Options

• v₁ represents a scalar multiple of A.
• v₂ represents a scalar multiple of A.
• v₅ represents a scalar multiple of B.
• v₄ represents a scalar multiple of A + B.

Concepts Explained 💡

• Scalar Multiple: The vector k * A is a scalar multiple of A. Geometrically, it lies on the same line as A (passing through the origin). If k > 0, it’s in the same direction; if k < 0, it’s in the opposite direction.
• Vector Addition: The vector A + B is found by adding the components. Geometrically, it is the diagonal of the parallelogram formed by vectors A and B.

Step-by-Step Solution

1. Analyze Scalar Multiples of A: Vector A = (-1, 2) is in the second quadrant. Any scalar multiple of A must lie on the line passing through the origin and (-1, 2). Both v₁ and v₂ lie on this line.
2. Analyze Scalar Multiples of B: Vector B = (2, -2) is in the fourth quadrant. Any scalar multiple of B must lie on the line passing through the origin and (2, -2). Vector v₅ lies on this line.
3. Analyze A + B: A + B = (-1, 2) + (2, -2) = (1, 0). This is a vector pointing along the positive x-axis. Any scalar multiple of A + B must lie on the x-axis. Vector v₄ lies on the positive x-axis.

Question 9 & 10: Solving a Vector Equation

Problem Let A = (1, 1, 1) and B = (2, -1, 4) be two vectors. Suppose c.A + 3B = (4, j, k), where c, j, k are real numbers (scalars). 9) Find the value of c. 10) Find the value of j + k.

Answers

• 9) c = -2
• 10) j + k = 5

Concepts Explained 💡

To solve a vector equation, perform the scalar multiplication and vector addition on one side. Then, equate the corresponding components of the vectors on both sides of the equation to form a system of simple equations.

Step-by-Step Solution

1. Set up the equation: c * (1, 1, 1) + 3 * (2, -1, 4) = (4, j, k)

2. Perform the operations on the left side: (c, c, c) + (6, -3, 12) = (4, j, k) (c + 6, c - 3, c + 12) = (4, j, k)

3. Equate components to find c (for Question 9): The first components must be equal: c + 6 = 4 c = 4 - 6 = -2

4. Use c to find j and k (for Question 10):

   • Equate the second components: j = c - 3 = -2 - 3 = -5
   • Equate the third components: k = c + 12 = -2 + 12 = 10

5. Calculate j + k: j + k = -5 + 10 = 5