Skip to main content
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 4 - Laws of Sines and Cosines; VectorsProblem 7
Chapter 4, Problem 7

In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i, w = j

Verified step by step guidance
1
Identify the components of the vectors \( \mathbf{v} \) and \( \mathbf{w} \). Here, \( \mathbf{v} = 5\mathbf{i} = (5, 0) \) and \( \mathbf{w} = \mathbf{j} = (0, 1) \).
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: \[ \mathbf{v} \cdot \mathbf{w} = 5 \times 0 + 0 \times 1 \]
Calculate \( \mathbf{v} \cdot \mathbf{v} \) by multiplying the components of \( \mathbf{v} \) with themselves and adding: \[ \mathbf{v} \cdot \mathbf{v} = 5 \times 5 + 0 \times 0 \]
Simplify the expressions from steps 3 and 4 to find the dot products.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
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 and w, v⋅w = v₁w₁ + v₂w₂ + ... + vₙwₙ.
Recommended video:
05:40
Introduction to Dot Product

Unit Vectors i and j

In two-dimensional space, i and j are standard unit vectors along the x-axis and y-axis, respectively. Vector i = (1, 0) and vector j = (0, 1). They are orthogonal, meaning their dot product is zero.
Recommended video:
06:01
i & j Notation

Dot Product of a Vector with Itself

The dot product of a vector with itself, v⋅v, equals the square of its magnitude. It is calculated by summing the squares of its components, which helps find the length or magnitude of the vector.
Recommended video:
05:40
Introduction to Dot Product
Related Practice
Textbook Question

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. C = 50°, a = 3, c = 1

2
views
1
rank
Textbook Question

In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

Textbook Question

In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

2
views
Textbook Question

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = i - j

Textbook Question
In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle.A = 162°, b = 11.2, c = 48.2
5
views
Textbook Question

In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. projᵥᵥv