Scalar Product Questions

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Very Short Answers Questions [1 Mark Questions]

Ques. The scalar product is also known as ______.

1. Dot product
2. Inner product
3. Cross product
4. Both a and b

Ans. The correct answer is d. Both a and b

Explanation: The scalar product is also known as the Dot product and Inner product. An inner product is a generalization of the dot product.

Ques. The scalar product gives the way of _______of two different vectors.

1. Subtraction
2. Addition
3. Multiplication
4. Division

Ans. The correct answer is c. Multiplication

Explanation: The scalar product gives the way of multiplication of two different vectors.

Ques. Scalar multiplication is denoted by a ______.

1. Dot
2. Comma
3. Semicolon
4. Dash

Ans. The correct answer is a. Dot

Explanation: The scalar multiplication of two vectors is represented by a Dot sign. For example, the scalar product of vector A and vector B is represented by \(\vec{A}.\vec{B}\)

Ques. Which among the following is a vector quantity?

1. Displacement
2. Mass
3. Density
4. Temperature

Ans. The correct answer is a. Displacement

Explanation: A vector quantity is a physical quantity that has both magnitude and direction. Among the following quantities in the option, only displacement has both magnitude and direction. Hence, it is a vector quantity.

Ques. Is electric charge a scalar quantity or vector quantity?

Ans. Electric charge is a scalar quantity because it has only magnitude but no sense of direction.

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Short Answers Questions [2 Marks Questions]

Ques. What is the formula to represent the magnitude of the scalar product?

Ans. Consider two vectors A and B, and θ be the angle between these two vectors. Then the formula to represent the magnitude of the scalar product of vectors A and B is given by

\(\vec{A}.\vec{B}\) = AB cosθ

Where

• A is the magnitude of vector A
• B is the magnitude of vector B
• Cosθ is the cosine of the angle between vectors A and B.

Ques. What is the formula to represent the magnitude of the vector product?

Ans. Consider two vectors A and B, and θ be the angle between these two vectors. Then the formula to represent the magnitude of the vector product of vectors A and B is given by

\(\vec{A} \times \vec{B}\) = AB sinθ

Where

• A is the magnitude of vector A
• B is the magnitude of vector B
• sinθ is the sine of the angle between vectors A and B.

Ques. Define the Scalar product of two vectors.

Ans. The scalar product of two vectors \(\vec{A}\) and \(​​\vec{B}\) denoted as \(\vec{A}.\vec{B}\) and is given by the product of the magnitude of both vectors multiplied by the cosine of the angle between them. i.e.

\(\vec{A}.\vec{B}\) = AB cosθ

Ques. Define the vector product of two vectors.

Ans. The vector product of two vectors \(\vec{A}\) and \(​​\vec{B}\) denoted as \(\vec{A} \times \vec{B}\) and is given by the product of the magnitude of both vectors multiplied by the sine of the angle between them. i.e.

\(\vec{A} \times \vec{B}\) = AB cosθ

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Long Answers Questions [3 Marks Questions]

Ques. State any four properties of the scalar product of two vectors.

Ans. The following are the properties of the scalar product of two vectors

• Scalar product is commutative i.e.\(\vec{A}.\vec{B}\) = \(\vec{B}.\vec{A}\)
• Scalar product is distributive i.e. \(\vec{A}.(\vec{B}+\vec{C}) = \vec{A}.\vec{B} + \vec{A}.\vec{C}\)
• Dot product of a vector with itself gives the square of its magnitude.
• If two vectors are parallel, then \(\vec{A}.\vec{B}\) = AB cos0 = AB

Ques. Let there be two vectors |a| = 6 and |b| = 2 and θ = 60°. Find their scalar product.

Ans. Given

• The magnitude of the first vector, |a| = 6
• The magnitude of the second vector, |b| = 2
• The angle between the two vectors, θ = 60°

The scalar product of vectors a and b is given by

\(\vec{a}.\vec{b}\) = abcos

⇒ \(\vec{a}.\vec{b}\) = 6 x 2 x cos60

⇒ \(\vec{a}.\vec{b}\) = 12cos60 = 12 x \(\frac{1}{2}\) = 6

Ques. If \(\vec{a}\) and \(\vec{b}\) are perpendicular vectors, \(|\vec{a} + \vec{b}|\) = 13 and \(|\vec{a}|\) = 5. Find the value of \(|\vec{b}|\).

Ans. Given \(|\vec{a} + \vec{b}|\) = 13

Squaring both sides we get

\(|\vec{a} + \vec{b}|\)² = 169

⇒ \(|\vec{a}|\)² + \(|\vec{b}|\)² + 2\(|\vec{a}|\)\(|\vec{b}|\)cos = 169

Since, both the vectors are perpendicular to each other, therefore, θ = 90

⇒ \(|\vec{a}|\)² + \(|\vec{b}|\)² + 2\(|\vec{a}|\)\(|\vec{b}|\) cos90 = 169

⇒ \(|\vec{a}|\)² + \(|\vec{b}|\)² + 2\(|\vec{a}|\)\(|\vec{b}|\) x 0 = 169

Given \(|\vec{a}|\) = 5

⇒ 5² + \(|\vec{b}|\)² = 169

⇒ \(|\vec{b}|\)² = 169 – 25 = 144

⇒ \(|\vec{b}|\) = 12

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Very Long Answers Questions [5 Marks Questions]

Ques. If the magnitude of two vectors is 4 and 6 and the magnitude of the scalar product is 12√2. What is the angle between the vectors?

Ans. Let the two vectors be A and B. Then it is given

• A = 4
• B = 6
• The magnitude of the scalar product, \(\vec{A}.\vec{B}\) = 12√2

Let θ be the angle between the vectors A and B.

We have, \(\vec{A}.\vec{B}\) = AB cosθ

⇒ cosθ = \(\frac{\vec{A}.\vec{B}}{AB}\)

⇒ cosθ = 12√2/(4 x 6) = 1/√2

⇒ θ = cos^-1 1/√2 = 45°

Ques. Find the angle between the force \(\vec{F}\) = \((5\hat{i} + 4\hat{j} + 5\hat{k})\) unit and displacement \(\vec{d}\) = \((3\hat{i} + 4\hat{j} - 3\hat{k})\) unit. Also, find the projection of the F vector on the d vector.

Ans. The magnitude of the F vector is given by

F = √(5² + 4² + 5²) = √66 units

The magnitude of the d vector is given by

d = √(3² + 4² – 3²) = √34 units

The scalar product of F and d is given by

\(\vec{F}\).\(\vec{d}\) = \((5\hat{i} + 4\hat{j} + 5\hat{k})\) . \((3\hat{i} + 4\hat{j} - 3\hat{k})\) = 16 unit

We have, \(\vec{F}\).\(\vec{d}\) = Fd cosθ

⇒ cosθ = \(\frac{\vec{F}.\vec{d}}{Fd}\)

⇒ cosθ = 16 /(√66 x √34) = 0.34

Now, projection of F on d = F cosθ = √66 x 0.34 = 2.76

Ques. Find the scalar and vector products of two vectors \(\vec{A} = (3\hat{i} – 4\hat{j} + 5\hat{k})\) and \(\vec{B} = (– 2\hat{i} + \hat{j} – 3\hat{k})\).

Ans. The scalar product of vectors A and B is given by

\(\vec{A}.\vec{B} = (3\hat{i} – 4\hat{j} + 5\hat{k}). (-2\hat{i}+\hat{j}-3\hat{k})\)

⇒\(\vec{A}.\vec{B}\) = – 6 – 4 – 15 = – 25

The vector product of vectors A and B is given by

\(\vec{A} \times \vec{B} = (3\hat{i} – 4\hat{j} + 5\hat{k}) \times (-2\hat{i}+\hat{j}-3\hat{k})\)

⇒ \(\vec{A} \times \vec{B}\) \(= (7\hat{i} – \hat{j} – 5\hat{k})\)

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