Find a⃗ ·b⃗ and the angle between a⃗ and b⃗ when they are tail-to-tail. a⃗=i⃗+3 j⃗-5 k⃗

Name: Problem 18, Chapter 10: Precalculus with Trigonometry: Concepts and Applications
Uploaded: 2021-09-18T13:59:25-08:00
Duration: 1 min 35 s
Channel: Yujie Wang
Description: Find a⃗ ·b⃗ and the angle between a⃗ and b⃗ when they are tail-to-tail. a⃗=i⃗+3 j⃗-5 k⃗ b⃗=-7 i⃗+4 j⃗+k⃗

step 1 Okay, so vector A and vector B. So let's first find A .B, okay? I just multiply the correspondin

Question

Find $\vec{a} \cdot \vec{b}$ and the angle between $\vec{a}$ and $\vec{b}$ when they are tail-to-tail. $$\begin{aligned} $$\begin{aligned} &\vec{a}=\vec{i}+3 \vec{j}-5 \vec{k}\\ &\vec{b}=-7 \vec{i}+4 \vec{j}+\vec{k} \end{aligned}$$

   Find $\vec{a} \cdot \vec{b}$ and the angle between $\vec{a}$ and $\vec{b}$ when they are tail-to-tail.
$$\begin{aligned}
$$\begin{aligned}
&\vec{a}=\vec{i}+3 \vec{j}-5 \vec{k}\\
&\vec{b}=-7 \vec{i}+4 \vec{j}+\vec{k}
\end{aligned}$$

Precalculus with Trigonometry: Concepts and Applications

Paul A. Foerster 1st Edition

Chapter 10, Problem 18

Instant Answer

Solved by Expert Yujie Wang

09/18/2021

Step 1

The dot product of two vectors is calculated by multiplying their corresponding components and then adding them up. So, we have: \[\vec{a} \cdot \vec{b} = (1 \cdot -7) + (3 \cdot 4) + (-5 \cdot 1) = -7 + 12 - 5 = 0\] Show more…

Show all steps

Thanks for your feedback!

Find $\vec{a} \cdot \vec{b}$ and the angle between $\vec{a}$ and $\vec{b}$ when they are tail-to-tail. $$\begin{aligned} $$\begin{aligned} &\vec{a}=\vec{i}+3 \vec{j}-5 \vec{k}\\ &\vec{b}=-7 \vec{i}+4 \vec{j}+\vec{k} \end{aligned}$$

Play audio

Feedback

Yujie Wang and 88 other subject Calculus 3 educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Dot Product

The dot product is an algebraic operation that takes two vectors and returns a scalar. It is computed by multiplying corresponding components of the vectors and then summing the results. This operation is fundamental in relating the geometric properties of vectors, such as the angle between them.

Vector Magnitude

The magnitude of a vector, also known as its norm, is a measure of its length in space. It is calculated as the square root of the sum of the squares of its components, and is essential for normalizing vectors and computing angles between them.

Angle Between Vectors

The angle between two vectors is determined using the dot product formula, where the cosine of the angle is given by the ratio of the dot product of the vectors to the product of their magnitudes. This relationship allows one to quantify the directional difference between the vectors, especially when they are positioned tail-to-tail.

Please give Ace some feedback