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Chapter 3: Problem 43

Let \(A=15^{T}-40 \mathrm{J}\) and \(B=31 \mathrm{J}+18 \mathrm{k} .\) Find \(C\) such that \(\vec{A}+\vec{B}+\vec{C}=\overrightarrow{0}\)

Short Answer

Expert verified
The vector \( C \) such that the sum of vectors \( A, B, \) and \( C \) equals to the zero vector is \( C = -15i + 9j - 18k \)

Step by step solution

01

Identify Given Vectors

The vectors given in the problem are \( A = 15i - 40j \) and \( B = 31j + 18k \). The goal is to find the third vector \( C \) such that the sum of the three vectors is zero.
02

Vector Addition

Since the sum of the vectors \( A, B, \) and \( C \) is a zero vector, this can be written mathematically as \( A + B + C = 0 \). This equation can be rearranged to solve for \( C \) by subtracting \( A \) and \( B \) from both sides, which gives: \( C = -A - B \).
03

Substitute the Given Vectors

By substituting the given vectors \( A \) and \( B \) into the equation, we get: \( C = - (15i - 40j) - (31j + 18k) \).
04

Simplify the Expression

Simplifying the above equation gives: \( C = -15i + 40j - 31j - 18k \)
05

Combine Similar Terms

After combining similar terms, the final expression for vector \( C \) becomes: \( C = -15i + 9j - 18k \)

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