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: u000AFirst, calculate u005C(2 u005Cmathbf{v}u005C). Given u005C(u005Cmathbf{v} u003D u002D3 u005Cmathbf{i} + 3 u005Cmathbf{j} +

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. $$ \mathbf{u}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k} \quad \mathbf{v}=-3 \mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \quad \mathbf{w}=\mathbf{i}+\mathbf{j}+3 \mathbf{k} $$ $\mathbf{u} \times(2 \mathbf{v})$

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

Key Concepts

Vector Cross Product

The cross product is a binary operation on two vectors in three-dimensional space that results in another vector perpendicular to the plane containing the original vectors. Its magnitude corresponds to the area of the parallelogram spanned by the vectors, and its direction is determined using the right-hand rule. This operation is fundamental in physics and engineering for calculating torques, angular momentum, and other quantities that involve rotational behavior.

Scalar Multiplication in Vector Operations

Scalar multiplication involves multiplying a vector by a constant, which scales the magnitude of the vector without changing its direction (unless the scalar is negative, in which case the direction is reversed). When a scalar multiplies a vector that is part of another operation, such as the cross product, it can be factored out due to the linearity of vector operations. This property simplifies many calculations in vector algebra.

Linearity of the Cross Product

The cross product is linear, meaning it satisfies the distributive property over vector addition and is compatible with scalar multiplication. In expressions like u × (2v), the scalar 2 can be factored out as 2(u × v). This linearity is crucial for breaking down complex expressions into simpler parts and is central to many proofs and computational techniques in vector calculus and physics.

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