Question

$$\text { Calculate }(3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}) \cdot(\mathbf{i}+2 \mathbf{j}-\mathbf{k})$$

   $$\text { Calculate }(3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}) \cdot(\mathbf{i}+2 \mathbf{j}-\mathbf{k})$$

Vector Calculus

Vector Calculus

Jerrold E. Marsden,… 6th Edition

Chapter 1, Problem 1

Instant Answer

Solved by Expert Nick Johnson

07/11/2021

Step 1

So, we have: $$(3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}) \cdot(\mathbf{i}+2 \mathbf{j}-\mathbf{k}) = 3*1 + 2*2 + 1*(-1)$$ Show more…

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$$\text { Calculate }(3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}) \cdot(\mathbf{i}+2 \mathbf{j}-\mathbf{k})$$

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Key Concepts

Vectors

Vectors are mathematical entities defined by both magnitude and direction. In many applications, especially in physics and engineering, they are represented in component form using standard basis vectors, such as i, j, and k in three-dimensional space. This representation simplifies many operations, including calculating magnitudes and directions.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a scalar value. It is calculated by multiplying the corresponding components of the vectors and summing these products. This operation is fundamental in determining the angle between vectors and checking for orthogonality.

Component-wise Multiplication

Component-wise multiplication is the process of multiplying corresponding elements of the two vectors. When performing the dot product, you multiply each pair of corresponding components from the vectors and then add the resulting products together to obtain a single scalar value.

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