In Problems 51-58, find the dot product 𝐯 ·𝐰 and the angle between 𝐯 and 𝐰. 𝐯=𝐢+𝐣, 𝐰=-𝐢+𝐣-𝐤

Step 1: Calculate the dot product $u005Cmathbf{v} u005Ccdot u005Cmathbf{w}$.u000AThe vectors are given as $u005Cmath

In Problems 51-58, find the dot product 𝐯 ·𝐰 and the angle...

In Problems 51-58, find the dot product $\mathbf{v} \cdot \mathbf{w}$ and the angle between $\mathbf{v}$ and $\mathbf{w}$.
$\mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j}-\mathbf{k}$

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Key Concepts

Magnitude of a Vector

The magnitude (or norm) of a vector is a measure of its length in space. It is calculated as the square root of the sum of the squares of its components. The magnitude of vectors is essential when normalizing them or when determining the angle between vectors.

Angle Between Vectors Using the Dot Product

The angle between two vectors can be found using the dot product formula, which relates the dot product to the product of the magnitudes of the vectors and the cosine of the angle between them. Specifically, the formula cos(?) = (v • w) / (|v| |w|) allows one to compute the cosine of the angle, and subsequently the angle itself. This is a fundamental concept in understanding the geometric relationship between vectors.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is computed by multiplying corresponding components of the vectors and then summing those products. This operation plays a crucial role in determining how aligned two vectors are.

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