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CBSE Class 12 Math 2024 All Sets Solved Paper

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Question : 8 of 20
Marks: +1, -0
For any two vectors a⃗\vec{a} and b⃗\vec{b}, which of the following statements is always true ?
Solution:  
Let us assume a⃗≠0⃗\vec{a} \neq \vec{0} and b⃗≠0⃗\vec{b} \neq \vec{0}
a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta
Taking modulus on both sides then we get,
∣a⃗⋅b⃗∣=∣a⃗∣∣b⃗∣∣cos⁡θ∣|\vec{a} \cdot \vec{b}| = |\vec{a}| |\vec{b}| |\cos \theta|
∣cos⁡θ∣=∣a⃗⋅b⃗∣∣a⃗∣∣b⃗∣|\cos \theta| = \frac{|\vec{a} \cdot \vec{b}|}{|\vec{a}| |\vec{b}|}
As we know that 0≤∣cos⁡θ∣≤10 \leq |\cos \theta| \leq 1
∣a⃗⋅b⃗∣∣a⃗∣∣b⃗∣≤1\frac{|\vec{a} \cdot \vec{b}|}{|\vec{a}| |\vec{b}|} \leq 1
Therefore, ∣a⃗⋅b⃗∣≤∣a⃗∣∣b⃗∣|\vec{a} \cdot \vec{b}| \leq |\vec{a}| |\vec{b}|
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