Defining the cross product The cross product of two nonzero vectors

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YUU SIIULIU L a e yull SWOIS WILll lie PUSULU SULULIUS. Lavas, 2 Defining the cross product The cross product of two nonzero

Defining the cross product The cross product of two nonzero vectors $\vec{u}$ and $\vec{v}$ is another vector $\vec{u} \times \vec{v}$ with magnitude

$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$

where $0 \leq \theta \leq \pi$ is the angle between the two vectors. The direction of $\vec{u} \times \vec{v}$ is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from $\vec{u}$ to $\vec{v}$ the direction of $\vec{u} \times \vec{v}$ is the direction of your thumb, orthogonal to both $\vec{u}$ and $\vec{v}$.

Evaluating the cross product. Let $\mathbf{u}=u_{1} \mathbf{i}+u_{2} \mathbf{j}+u_{3} \mathbf{k}$ and $\mathbf{v}=v_{1} \mathbf{i}+v_{2} \mathbf{j}+v_{3} \mathbf{k}$. Then

$\mathbf{u} \times \mathbf{v}=\left|\begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3}\end{array}\right|=\left|\begin{array}{ll}u_{2} & u_{3} \\ v_{2} & v_{3}\end{array}\right| \mathbf{i}-\left|\begin{array}{ll}u_{1} & u_{3} \\ v_{1} & v_{3}\end{array}\right| \mathbf{j}+\left|\begin{array}{ll}u_{1} & u_{2} \\ v_{1} & v_{2}\end{array}\right| \mathbf{k}$

(1) Let $\vec{v}=\vec{i}+2 \vec{j}-3 \vec{k}$ and $\vec{w}=\vec{i}-2 \vec{j}+\vec{k}$. Find the following:

(a) $\overrightarrow{\vec{v} \cdot \vec{w}}$

(d) Find a unit vector orthogonal to both $\vec{v}$

(b) $(\vec{v} \cdot \vec{w}) \vec{w}$

and $\vec{w}$

(c) The angle between $\vec{v}$ and $\vec{w}$

(e) $\operatorname{proj}_{\vec{v}} \vec{w}$.

(2) Let $\vec{u}, \vec{v}$, and $\vec{w}$ be vectors. For each of the following expressions, identify whether the resulting value is a scalar, a vector, or the expression is not defined. If it is not defined, explain why. If a cross product is involved, assume the vectors are in three dimensions. If a division is involved, assume the denominator is nonzero.

(a) $\vec{u}^{2}$

(d) $\vec{u} \times(\vec{v} \cdot \vec{w})$

(g) $\frac{\vec{u} \times \vec{w}}{\vec{v} \cdot \vec{w}}$

(b) $|\vec{u}|^{2}$

(e) $|\vec{u}| \times|\vec{v}|$

(h) $\frac{\vec{u} \cdot \vec{w}}{\vec{v} \times \vec{w}}$

(c) $(\vec{u} \times \vec{v}) \cdot \vec{w}$

(f) $\frac{\vec{u} \cdot \vec{w}}{\vec{v} \cdot \vec{w}}$

math calculus

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