Find the scalar product (a.b) and the vector product (𝐚 ×𝐛), when (a) 𝐚=𝐢+2 𝐣-𝐤, 𝐛=2 𝐢+3 𝐣+𝐤 (b)

Name: Problem 7, Chapter 18: Engineering Mathematics
Uploaded: 2022-01-11T16:10:57-08:00
Duration: 3 min 14 s
Channel: Mkhitar Hobosyan
Description: Find the scalar product (a.b) and the vector product (𝐚 ×𝐛), when (a) 𝐚=𝐢+2 𝐣-𝐤, 𝐛=2 𝐢+3 𝐣+𝐤 (b) 𝐚=2 𝐢+3 𝐣+4 𝐤, 𝐛=5 𝐢-2 𝐢+𝐤

step 1 Hi, we need to find the scholar product and the vector product of the given vectors. Let's start

Question

Find the scalar product (a.b) and the vector product $(\mathbf{a} \times \mathbf{b})$, when (a) $\mathbf{a}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}, \mathbf{b}=2 \mathbf{i}+3 \mathbf{j}+\mathbf{k}$ (b) $\mathbf{a}=2 \mathbf{i}+3 \mathbf{j}+4 \mathbf{k}, \mathbf{b}=5 \mathbf{i}-2 \mathbf{i}+\mathbf{k}$

    Find the scalar product (a.b) and the vector product $(\mathbf{a} \times \mathbf{b})$, when
(a) $\mathbf{a}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}, \mathbf{b}=2 \mathbf{i}+3 \mathbf{j}+\mathbf{k}$
(b) $\mathbf{a}=2 \mathbf{i}+3 \mathbf{j}+4 \mathbf{k}, \mathbf{b}=5 \mathbf{i}-2 \mathbf{i}+\mathbf{k}$

Engineering Mathematics

K. A. Stroud, Dexter… 5th Edition

Chapter 18, Problem 7

Instant Answer

Solved by Expert Mkhitar Hobosyan

01/11/2022

Step 1

The scalar product (also known as the dot product) of two vectors $\mathbf{a}$ and $\mathbf{b}$ is given by the formula $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$. Show more…

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Find the scalar product (a.b) and the vector product $(\mathbf{a} \times \mathbf{b})$, when (a) $\mathbf{a}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}, \mathbf{b}=2 \mathbf{i}+3 \mathbf{j}+\mathbf{k}$ (b) $\mathbf{a}=2 \mathbf{i}+3 \mathbf{j}+4 \mathbf{k}, \mathbf{b}=5 \mathbf{i}-2 \mathbf{i}+\mathbf{k}$

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Key Concepts

Dot Product

The dot product, or scalar product, provides a scalar value calculated by summing the products of corresponding components of two vectors. It measures the extent to which two vectors point in the same direction and is mathematically expressed as the sum of the products of their corresponding components.

Cross Product

The cross product, or vector product, results in a vector that is perpendicular to the plane defined by the two input vectors. Its magnitude is equal to the product of the magnitudes of the original vectors times the sine of the angle between them, and its direction is given by the right-hand rule.

Determinant Method for Cross Product

The determinant method uses a 3x3 matrix with unit vectors in the first row and the components of the two vectors in the subsequent rows. By calculating the determinant using cofactor expansion, one systematically finds the components of the cross product vector.

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