Short Quiz - Chapter 10 Class 12 Vector Algebra

Correct option: B

Answer: Given:

$$ (\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=198 $$


Use identity:

$$ (\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})=|\vec{a}|^2-|\vec{b}|^2 $$


Also:

$$ |\vec{a}|=10|\vec{b}| $$


Let:

$$ |\vec{b}|=x $$


Then:

$$ |\vec{a}|=10 x $$


So:

$$ \begin{gathered} (10 x)^2-x^2=198 \\ 100 x^2-x^2=198 \\ 99 x^2=198 \\ x^2=2 \\ x=\sqrt{2} \end{gathered} $$


Therefore:

$$ |\vec{b}|=\sqrt{2} $$


Answer: B) \(|\vec{b}|=\sqrt{2}\)

Correct option: A

Answer: For any two vectors:

$$ \vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta $$


Since:

$$ \cos \theta \leq 1 $$


Therefore:

$$ \vec{a} \cdot \vec{b} \leq|\vec{a}||\vec{b}| $$


This is always true.
Answer: A) \(\vec{a} \cdot \vec{b} \leq|\vec{a}||\vec{b}|\)

Correct option: A

Answer: Assertion (A): The vectors \(a\) and ( \(-2 a\) ), where \(a \neq 0\), are collinear vectors.
Reason (R): \(a \cdot(-2 a)=0\).
Step-by-Step Reasoning:
1. Analyze Assertion (A): Two vectors are collinear if one can be expressed as a scalar multiple of the other ( \(u=k v\) ). Since ( \(-2 a\) ) is simply -2 times the vector \(a\), they are by definition collinear (they lie on the same or parallel lines). Therefore, Assertion (A) is True.
2. Analyze Reason (R): Let's calculate the dot product \(a \cdot(-2 a)\) :

$$ a \cdot(-2 a)=-2(a \cdot a)=-2|a|^2 $$


Since we are given that \(a \neq 0,|a|^2\) must be greater than zero. Therefore, the dot product is \(-2|a|^2\), which is not equal to 0 . The only way a dot product is zero is if the vectors are perpendicular (orthogonal), which these are not. Therefore, Reason (R) is False.

Answer: C) A is true, but R is false.

Correct option: D

Answer: Assertion (A): Projection of \(a\) on \(b\) is the same as the projection of \(b\) on \(a\).
Reason (R): The angle between \(a\) and \(b\) is the same as the angle between \(b\) and \(a\) numerically.
Step-by-Step Reasoning:
1. Analyze Assertion (A): The formula for the scalar projection of vector \(a\) onto vector \(b\) is \(\frac{a \cdot b}{|b|}\). The formula for the projection of \(b\) onto \(a\) is \(\frac{b \cdot a}{|a|}\). These are only equal if \(|a|=|b|\). In the general case where the magnitudes of the two vectors are different, the projections are not the same. Therefore, Assertion (A) is False.
2. Analyze Reason (R): The angle between two vectors and b is defined in the same plane and is numerically identical regardless of the order in which you name the pair. Therefore, Reason (R) is True.

Answer: D) A is false, but R is true.