Use the vectors 𝐮=2 𝐢+𝐣, 𝐯=-3 𝐢-10 𝐣, and 𝐰=𝐢-5 𝐣 Perform the indicated vector operations

Name: Problem 45, Chapter 8: Algebra and Trigonometry
Uploaded: 2020-07-06T07:10:21-08:00
Duration: 1 min 48 s
Channel: Ankit Gupta
Description: Use the vectors 𝐮=2 𝐢+𝐣, 𝐯=-3 𝐢-10 𝐣, and 𝐰=𝐢-5 𝐣 Perform the indicated vector operations and state the answer in two forms: (a) as a linear combination of 𝐢 and j and (b) in component form. 4 𝐮-5 𝐰

step 1 The first part of the question is asking about the linear combination of I and Z and the questio

Question

Use the vectors $\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{v}=-3 \mathbf{i}-10 \mathbf{j}, \quad$ and $\quad \mathbf{w}=\mathbf{i}-5 \mathbf{j}$ Perform the indicated vector operations and state the answer in two forms: (a) as a linear combination of $\mathbf{i}$ and j and (b) in component form. $4 \mathbf{u}-5 \mathbf{w}$

   Use the vectors
$\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{v}=-3 \mathbf{i}-10 \mathbf{j}, \quad$ and $\quad \mathbf{w}=\mathbf{i}-5 \mathbf{j}$
Perform the indicated vector operations and state the answer in two forms: (a) as a linear combination of $\mathbf{i}$ and j and (b) in component form.

$4 \mathbf{u}-5 \mathbf{w}$

Algebra and Trigonometry

Judith A. Beecher,… 4th Edition

Chapter 8, Problem 45

Instant Answer

Solved by Expert Ankit Gupta

07/06/2020

Step 1

So, we have $4(2\mathbf{i}+\mathbf{j})-5(\mathbf{i}-5\mathbf{j})$. Show more…

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Use the vectors $\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{v}=-3 \mathbf{i}-10 \mathbf{j}, \quad$ and $\quad \mathbf{w}=\mathbf{i}-5 \mathbf{j}$ Perform the indicated vector operations and state the answer in two forms: (a) as a linear combination of $\mathbf{i}$ and j and (b) in component form. $4 \mathbf{u}-5 \mathbf{w}$

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Key Concepts

Linear Combination

A linear combination is an expression constructed from a set of terms by multiplying each term by a constant and then adding the results. In the context of vectors, it means creating a new vector by scaling and summing other vectors, which underpins many operations in linear algebra.

Vector Arithmetic

Vector arithmetic involves operations such as addition and scalar multiplication, which are performed componentwise. This concept is fundamental in understanding how to manipulate vectors in any vector space by combining and scaling them appropriately.

Component and Basis Representation

Vectors can be expressed in terms of their components relative to a set of basis vectors (for example, i and j in the plane). The basis representation (linear combination of i and j) makes explicit the contribution of each standard direction, while the component representation provides a concise ordered pair that lists the scalar values along each coordinate axis.

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