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The vectors $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ satisfy $\|\mathbf{a}\| = \|\mathbf{b}\| = 1,$ $\|\mathbf{c}\| = 2,$ and \[\mathbf{a} \times (\mathbf{a} \times \mathbf{c}) + \mathbf{b} = \mathbf{0}.\]If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{c},$ then find all possible values of $\theta,$ in degrees.

 

What I tried: 

I began with subtracting both sides with b, and taking the magnitude.

I get \[|\mathbf{a} \times (\mathbf{a} \times \mathbf{c})|=1\]

\(|a|*|c|*\sin(\theta)*\sin(\theta)=1\)

\(\sin^2(\theta)=1/2, \sin(\theta)=\dfrac{1}{\sqrt2}\)

\(\theta=45, 135, 225, 315\)

 

The answer registered it incorrect, can somebody help?

 Jun 7, 2022
 #1
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The angle for the second vector product is 90 deg not theta.

a cross c is perpendicular to a .

 Jun 8, 2022

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