Mathematics Exam Questions and Answers for Class 12 on “Multiplication of a Vector by a Scalar”.
1. Multiplication of vector (vec{a}) and scalar λ is denoted as ______
a) λ(vec{a})
b) (vec{a})
c) λ
d) 0
Answer: a
Clarification: Multiplication of vector (vec{a}) and scalar λ is denoted as λ(vec{a}), as (vec{a}) is the original vector. λ is the scalar which can have any integer value which is to be multiplied to the given vector(vec{a}), whereas 0 can only be the answer if the scalar λ = 0.
2. Direction of λ(vec{a}) and (vec{a}) is same if λ is _______
a) imaginary
b) negative
c) positive
d) zero
Answer: c
Clarification: Direction of λ(vec{a}) and (vec{a}) is same if value λ is positive as it gives it a direction which is positive in nature. If the value of λ is negative then the direction of the result after multiplication becomes in opposite direction. Whereas the value of the product vector becomes zero if value of λ is 0.
3. Find magnitude (vec{a}) =(hat{i}) + (hat{j}) + (hat{k}).
a) (sqrt{3})
b) (sqrt{2})
c) 0
d) (sqrt{4})
Answer: a
Clarification: Magnitude of vector is calculated by formula (sqrt{x^2+ y^2+ z^2}).
Where x, y, z are the coefficients of (hat{i}), (hat{j}), (hat{k}).
The magnitude of vector (vec{a}) is calculated as (sqrt{(1^2+1^2+1^2)} = sqrt{3}).
4. |λ| times the magnitude of vector (vec{a}) is denoted as ______
a) |λ(vec{a})|
b) λ|(vec{a})|
c) |λ|(vec{a})
d) λ(vec{a})
Answer: a
Clarification: |λ| times the magnitude of vector (vec{a}) is denoted as |λ(vec{a})| = |λ||(vec{a})|
As we know that the magnitude of vector (vec{a}) is denoted by |(vec{a})|, if we multiply the magnitude of vector (vec{a}) with magnitude of λ we get |λ(vec{a})|.
5. If (vec{a}) =(hat{i}) + (hat{j}) + (hat{k}) and λ=5, what is value of λ(vec{a})?
a) (hat{i}) + (hat{j}) + (hat{k})
b) 5(hat{i}) + 5(hat{j}) + 5(hat{k})
c) (hat{i}) + 5(hat{j}) + 5(hat{k})
d) 10(hat{i}) + 10(hat{j}) + 10(hat{k})
Answer: b
Clarification: Multiplication of vector (vec{a}) =(hat{i}) + (hat{j}) + (hat{k}) by scalar value 5 results in 5(hat{i}) + 5(hat{j}) + 5(hat{k}), as in these type of questions we multiply(hat{i}), (hat{j,})
(hat{k}) with the constant given and the answer comes out to be 5(hat{i}) + 5(hat{j}) + 5(hat{k}).
6. If k is any scalar and (vec{a}), (vec{b}) be vectors then k ((vec{a})+ (vec{b}))= ________
a) k(vec{a}) + k(vec{b})
b) k(vec{a}) + (vec{b})
c) (vec{a}) + k(vec{b})
d) (vec{a}) + (vec{b})
Answer: a
Clarification: Multiplication of vector by scalar satisfies distributive property over addition and in k ((vec{a})+ (vec{b})) we multiply k with (vec{a}), (vec{b}) individually and hence the answer comes out to be k(vec{a}) + k(vec{b}).
7. Find values of x, y, z if vectors (vec{a})=x(hat{i}) + 2(hat{j}) + z(hat{k}) and (vec{b})=2(hat{i}) + y(hat{j}) + (hat{k}) are equal.
a) x=2, y=2, z=1
b) x=1, y=2, z=1
c) x=2, y=1, z=1
d) x=2, y=2, z=2
Answer: a
Clarification: As both the vectors are equal hence, we can equate their constants and get the value of x, y and z. Now we equate the coefficients of (hat{i}), (hat{j}), (hat{k}) of both the equations and get the values x=2, y=2, z=1.
8. (vec{a})=(hat{i}) + 2(hat{j}) and (vec{b})=2(hat{i}) + (hat{j}) , Is |(vec{a})| = |(vec{b})|?
a) Yes
b) No
Answer: a
Clarification: As we know that magnitude of vector is calculated by formula (sqrt{x^2+ y^2}).
Therefore, |(vec{a})| = (sqrt{12} + 22 = sqrt{5}) and (|vec{b}| = sqrt{22} + 12 = sqrt{5}), they are equal.
9. What is direction of vector (vec{a}) if it is multiplied with -λ?
a) Downwards
b) Upwards
c) Same
d) Opposite
Answer: d
Clarification: If the vector is multiplied with –λ then its direction become opposite as the direction in which it was previous may be positive or negative. After it is multiplied with a negative value then its direction becomes exactly opposite to the previous direction.
10. If k is any scalar and (vec{a}), (vec{b}) be vectors then k (vec{a}) + m(vec{a}) can also be written as ________
a) (k+m)(vec{a})
b) (vec{a}) + m(vec{a})
c) k (vec{a}) + (vec{a})
d) mk(vec{a})
Answer: a
Clarification: It satisfies distribution property over addition, hence in k (vec{a}) + m(vec{a}) we can take the vector (vec{a})
common and the answer come out to be (k+m)(vec{a}). Basically it’s a simplification method by which the vectors can be easily solved and further properties can be applied to them.