In Problems 23-44, use the given vectors 𝐮, 𝐯, and 𝐰 to find each expression. 𝐮=2 𝐢-3 𝐣+𝐤 𝐯=-3

Step 1: Calculate u005C(u002D3 u005Cmathbf{v}u005C).u000AGiven u005C(u005Cmathbf{v} u003D u002D3 u005Cmathbf{i} + 3 u005Cmathbf{j} + 2 u005Cmath

In Problems 23-44, use the given vectors 𝐮, 𝐯, and 𝐰 to find...

In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression. $$ \mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k} $$ $(-3 \mathbf{v}) \times \mathbf{w}$

In Problems 23-44, use the given vectors $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ to find each expression.
$$
\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}
$$
$(-3 \mathbf{v}) \times \mathbf{w}$

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Key Concepts

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying each component of a vector by a scalar. This operation scales the vector's magnitude without changing its direction, unless the scalar is negative, in which case the direction is reversed. It is a fundamental operation in vector algebra that is used to adjust vector sizes before further operations are carried out.

Cross Product of Vectors

The cross product is a binary operation on two vectors in three-dimensional space that results in another vector which is perpendicular to the plane containing the original vectors. Its magnitude is equal to the product of the magnitudes of the two original vectors times the sine of the angle between them, and it is essential for finding directions orthogonal to a given plane and determining areas of parallelograms spanned by the vectors.

Linearity of the Cross Product

The cross product is a linear operation with respect to scalar multiplication and vector addition. This means that when a vector is scaled before performing the cross product, the scaling factor can be factored out of the cross product operation. This property simplifies computations by allowing scalar factors to be managed separately from the vector cross product operation.

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