In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 3i + j, w = i + 3j

Verified step by step guidance

Identify the components of the vectors \( \mathbf{v} = 3\mathbf{i} + \mathbf{j} \) and \( \mathbf{w} = \mathbf{i} + 3\mathbf{j} \). Here, \( \mathbf{v} = (3, 1) \) and \( \mathbf{w} = (1, 3) \).

Recall the formula for the dot product of two vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \): \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \]

Calculate \( \mathbf{v} \cdot \mathbf{w} \) by multiplying corresponding components and adding the results: \[ \mathbf{v} \cdot \mathbf{w} = 3 \times 1 + 1 \times 3 \]

Recall that \( \mathbf{v} \cdot \mathbf{v} \) is the dot product of \( \mathbf{v} \) with itself, which gives the square of its magnitude: \[ \mathbf{v} \cdot \mathbf{v} = 3 \times 3 + 1 \times 1 \]

Evaluate the sums from steps 3 and 4 to find the values of \( \mathbf{v} \cdot \mathbf{w} \) and \( \mathbf{v} \cdot \mathbf{v} \) respectively.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Dot Product of Vectors

The dot product is an algebraic operation that takes two vectors and returns a scalar. It is calculated by multiplying corresponding components of the vectors and summing the results. For vectors v = (v1, v2) and w = (w1, w2), the dot product is v⋅w = v1w1 + v2w2.

Vector Components and Notation

Vectors in two dimensions are often expressed in terms of unit vectors i and j, representing the x and y directions respectively. For example, v = 3i + j means the vector has components (3, 1). Understanding this notation is essential for performing operations like the dot product.

Properties of the Dot Product

The dot product is commutative, meaning v⋅w = w⋅v, and distributive over vector addition. Also, the dot product of a vector with itself, v⋅v, gives the square of its magnitude, which is useful for finding vector lengths and angles between vectors.

Recommended video:

05:40

Introduction to Dot Product

Related Practice

Textbook Question

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.sin 6x sin 2x

views

Textbook Question

Use the following conditions to solve Exercises 1–4:4 𝝅sin α = ----- , ------- < α < 𝝅5 25 𝝅cos β = ------ , 0 < β < ------13 2Find the exact value of each of the following.cos (α + β)

Textbook Question

In oblique triangle ABC, A = 34°, B = 68°, and a = 4.8. Find b to the nearest tenth.