The cross product between two vectors in ℝ^3 is the vector defined by the formula 𝐯 ×𝐰=( v2

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The cross product between two vectors in ℝ^3 is the vector...

The cross product between two vectors in $\mathbb{R}^3$ is the vector defined by the formula $$ \mathbf{v} \times \mathbf{w}=\left(\begin{array}{c} v_2 w_3-v_3 w_2 \\ v_3 w_1-v_1 w_3 \\ v_1 w_2-v_2 w_1 \end{array}\right), \quad \text { where }=\left(\begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array}\right), \quad \mathbf{w}=\left(\begin{array}{c} w_1 \\ w_2 \\ w_3 \end{array}\right) . $$ (a) Show that $\mathbf{u}=\mathbf{v} \times \mathbf{w}$ is orthogonal, under the dot product, to both $\mathbf{v}$ and $\mathbf{w}$. (b) Show that $\mathbf{v} \times \mathbf{w}=\mathbf{0}$ if and only if $\mathbf{v}$ and $\mathbf{w}$ are parallel. (c) Prove that if $\mathbf{v}, \mathbf{w} \in \mathbb{R}^3$ are orthogonal nonzero vectors, then $\mathbf{u}=\mathbf{v} \times \mathbf{w}, \mathbf{v}, \mathbf{w}$ form an orthogonal basis of $\mathbb{R}^3$. (d) True or false: If $\mathbf{v}, \mathbf{w} \in \mathbb{R}^3$ are orthogonal unit vectors, then $\mathbf{v}, \mathbf{w}$ and $\mathbf{u}=\mathbf{v} \times \mathbf{w}$ form an orthonormal basis of $\mathbb{R}^3$.

The cross product between two vectors in $\mathbb{R}^3$ is the vector defined by the formula
$$
\mathbf{v} \times \mathbf{w}=\left(\begin{array}{c}
v_2 w_3-v_3 w_2 \\
v_3 w_1-v_1 w_3 \\
v_1 w_2-v_2 w_1
\end{array}\right), \quad \text { where }=\left(\begin{array}{c}
v_1 \\
v_2 \\
v_3
\end{array}\right), \quad \mathbf{w}=\left(\begin{array}{c}
w_1 \\
w_2 \\
w_3
\end{array}\right) .
$$
(a) Show that $\mathbf{u}=\mathbf{v} \times \mathbf{w}$ is orthogonal, under the dot product, to both $\mathbf{v}$ and $\mathbf{w}$.
(b) Show that $\mathbf{v} \times \mathbf{w}=\mathbf{0}$ if and only if $\mathbf{v}$ and $\mathbf{w}$ are parallel. (c) Prove that if $\mathbf{v}, \mathbf{w} \in \mathbb{R}^3$ are orthogonal nonzero vectors, then $\mathbf{u}=\mathbf{v} \times \mathbf{w}, \mathbf{v}, \mathbf{w}$ form an orthogonal basis of $\mathbb{R}^3$.
(d) True or false: If $\mathbf{v}, \mathbf{w} \in \mathbb{R}^3$ are orthogonal unit vectors, then $\mathbf{v}, \mathbf{w}$ and $\mathbf{u}=\mathbf{v} \times \mathbf{w}$ form an orthonormal basis of $\mathbb{R}^3$.

Key Concepts

Cross Product

The cross product is an operation defined on two vectors in three-dimensional space that produces a new vector which is orthogonal to both of the original vectors. Its magnitude is given by the product of the magnitudes of the two vectors and the sine of the angle between them. This operation is fundamental in vector calculus and physics, particularly in determining orientations and directions perpendicular to a given plane.

Dot Product and Orthogonality

The dot product is a scalar product of two vectors that measures the extent to which the vectors point in the same direction. Two vectors are considered orthogonal (perpendicular) if their dot product equals zero. This concept is essential for proving properties involving perpendicularity, like showing that the result of a cross product is orthogonal to the original vectors.

Parallel Vectors

Vectors are parallel if they have the same or exactly opposite directions, which geometrically implies that one is a scalar multiple of the other. In the context of the cross product, if the vectors are parallel, then the sine of the angle between them is zero, leading to a zero vector as the result. This property serves as a diagnostic tool to determine if two vectors are parallel.

Basis and Linear Independence

A set of vectors forms a basis for a vector space if they are linearly independent and span the entire space. In three-dimensional space, three vectors that are mutually orthogonal are automatically linearly independent, and thus, if they are nonzero, they serve as an orthogonal basis. This concept is crucial when establishing coordinate systems and simplifying vector computations.

Orthonormal Basis

An orthonormal basis is an orthogonal basis consisting of unit vectors. This means that not only are all the vectors mutually perpendicular, but each vector has a length of one. Orthonormal bases are particularly useful in simplifying problems in both algebra and geometry, as they provide a convenient and standardized way of representing vectors in space.

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