If - and b- are two units vectors inclined at angle - to each other- then

A) nbsp;If | a + b| nbsp; lt; 1 then (a + b) · (a + b) lt; 1 so | a| ^2+| b| ^2 + 2a· b lt; 1 ⇒ a · b lt; 1/2 ⇒cosα lt; ...

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Standard XII

Mathematics

Applications of Dot Product

If â and ...

Question

If $^a$ and $^b$ are two units vectors inclined at angle $α$ to each other, then

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Solution

A) If $| a + b | < 1$ then $(a + b) \cdot (a + b) < 1$
so ${| a |}^{2} + {| b |}^{2} + 2 a \cdot b < 1 \Rightarrow a \cdot b < - 1 / 2$
$\Rightarrow cos α < - 1 / 2 \Rightarrow 2 π / 3 < α < π$ .
B) If $| a - b | = | a + b |$ then $a \cdot b = 0 \Rightarrow α = π / 2$
C) If $| a + b | < \sqrt{2}$ then $cos θ < 0$ which is true if $π / 2 < θ π$
D) If $| a + b | < \sqrt{2}$ then $cos θ > 0$ which is true if $0 \leq θ < π / 2$

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If ^a and ^b are two units vectors inclined at angle α to each other, then

A) If |a+b|<1 then (a+b)⋅(a+b)<1so |a|2+|b|2+2a⋅b<1⇒a⋅b<−1/2⇒cosα<−1/2⇒2π/3<α<π.B) If |a−b|=|a+b| then a⋅b=0⇒α=π/2C) If |a+b|<√2 then cosθ<0 which is true if π/2<θπD) If |a+b|<√2 then cosθ>0 which is true if 0≤θ<π/2

If $^a$ and $^b$ are two units vectors inclined at angle $α$ to each other, then