Two unit vectors a-hat and b-hat satisfy | $\hat{a}$ + $\hat{b}$ |=1. Which set is correct for $\hat{a}$ · $\hat{b}$ , | $\hat{a}$ × $\hat{b}$ | and | $\hat{a}$ - $\hat{b}$ |?
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Question

Two unit vectors a-hat and b-hat satisfy | $\hat{a}$ + $\hat{b}$ |=1. Which set is correct for $\hat{a}$ · $\hat{b}$ , | $\hat{a}$ × $\hat{b}$ | and | $\hat{a}$ - $\hat{b}$ |?

Options

G1 Physics · Accepted Answer

1=|a+b|²=2+2a·b, so a·b=-1/2 and θ=120°. Cross magnitude=sin120°=√3/2. Difference squared=2-2(-1/2)=3, so |a-b|=√3.

Two unit vectors a-hat and b-hat satisfy | a ^ \hat{a} a ^ + b ^ \hat{b} b ^ |=1. Which set is correct for a ^ \hat{a} a ^ · b ^ \hat{b} b ^ , | a ^ \hat{a} a ^ × b ^ \hat{b} b ^ | and | a ^ \hat{a} a ^ - b ^ \hat{b} b ^ |?