Unit vector in the direction of $\left\langle 3 \sin^{2}{\left(t \right)} \cos{\left(t \right)}, - 3 \sin{\left(t \right)} \cos^{2}{\left(t \right)}, \sin{\left(2 t \right)}\right\rangle$

The calculator will find the unit vector in the direction of the vector $\left\langle 3 \sin^{2}{\left(t \right)} \cos{\left(t \right)}, - 3 \sin{\left(t \right)} \cos^{2}{\left(t \right)}, \sin{\left(2 t \right)}\right\rangle$, with steps shown.
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Find the unit vector in the direction of $\mathbf{\vec{u}} = \left\langle 3 \sin^{2}{\left(t \right)} \cos{\left(t \right)}, - 3 \sin{\left(t \right)} \cos^{2}{\left(t \right)}, \sin{\left(2 t \right)}\right\rangle.$

Solution

The magnitude of the vector is $\mathbf{\left\lvert\vec{u}\right\rvert} = \frac{\sqrt{26 - 26 \cos{\left(4 t \right)}}}{4}$ (for steps, see magnitude calculator).

The unit vector is obtained by dividing each coordinate of the given vector by the magnitude.

Thus, the unit vector is $\mathbf{\vec{e}} = \left\langle \frac{6 \sqrt{26} \sin^{2}{\left(t \right)} \cos{\left(t \right)}}{13 \sqrt{1 - \cos{\left(4 t \right)}}}, - \frac{6 \sqrt{26} \sin{\left(t \right)} \cos^{2}{\left(t \right)}}{13 \sqrt{1 - \cos{\left(4 t \right)}}}, \frac{2 \sqrt{26} \sin{\left(2 t \right)}}{13 \sqrt{1 - \cos{\left(4 t \right)}}}\right\rangle$ (for steps, see vector scalar multiplication calculator).

The unit vector in the direction of $\left\langle 3 \sin^{2}{\left(t \right)} \cos{\left(t \right)}, - 3 \sin{\left(t \right)} \cos^{2}{\left(t \right)}, \sin{\left(2 t \right)}\right\rangle$A is $\left\langle \frac{6 \sqrt{26} \sin^{2}{\left(t \right)} \cos{\left(t \right)}}{13 \sqrt{1 - \cos{\left(4 t \right)}}}, - \frac{6 \sqrt{26} \sin{\left(t \right)} \cos^{2}{\left(t \right)}}{13 \sqrt{1 - \cos{\left(4 t \right)}}}, \frac{2 \sqrt{26} \sin{\left(2 t \right)}}{13 \sqrt{1 - \cos{\left(4 t \right)}}}\right\rangle\approx \left\langle \frac{2.353393621658208 \sin^{2}{\left(t \right)} \cos{\left(t \right)}}{\left(1 - \cos{\left(4 t \right)}\right)^{0.5}}, - \frac{2.353393621658208 \sin{\left(t \right)} \cos^{2}{\left(t \right)}}{\left(1 - \cos{\left(4 t \right)}\right)^{0.5}}, \frac{0.784464540552736 \sin{\left(2 t \right)}}{\left(1 - \cos{\left(4 t \right)}\right)^{0.5}}\right\rangle.$A