Question:
Find the sum of the vectors \(\vec{a}=\hat{i}-2\hat{j}+\hat{k},\vec{b}=-2\hat{i}+4\hat{j}+5\hat{k}\) and \(\vec{c}=\hat{i}-6\hat{j}-7\hat{k}.\)
Find the sum of the vectors \(\vec{a}=\hat{i}-2\hat{j}+\hat{k},\vec{b}=-2\hat{i}+4\hat{j}+5\hat{k}\) and \(\vec{c}=\hat{i}-6\hat{j}-7\hat{k}.\)
Updated On: Sep 19, 2023
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Solution and Explanation
The correct answer is:\(-4\hat{j}-\hat{k}\)
The given vectors are \(\vec{a}=\hat{i}-2\hat{j}+\hat{k},\vec{b}=-2\hat{i}+4\hat{j}+5\hat{k}\) and \(\vec{c}=\hat{i}-6\hat{j}-7\hat{k}.\)
\(∴\vec{a}+\vec{b}+\vec{c}=(1-2+1)\hat{i}+(-2+4-6)\hat{j}+(1+5-7)\hat{k}\)
\(=0.\hat{i}-4\hat{j}-1.\hat{k}\)
\(=-4\hat{j}-\hat{k}\)
The given vectors are \(\vec{a}=\hat{i}-2\hat{j}+\hat{k},\vec{b}=-2\hat{i}+4\hat{j}+5\hat{k}\) and \(\vec{c}=\hat{i}-6\hat{j}-7\hat{k}.\)
\(∴\vec{a}+\vec{b}+\vec{c}=(1-2+1)\hat{i}+(-2+4-6)\hat{j}+(1+5-7)\hat{k}\)
\(=0.\hat{i}-4\hat{j}-1.\hat{k}\)
\(=-4\hat{j}-\hat{k}\)
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Concepts Used:
Multiplication of a Vector by a Scalar
When a vector is multiplied by a scalar quantity, the magnitude of the vector changes in proportion to the scalar magnitude, but the direction of the vector remains the same.
Properties of Scalar Multiplication:
The Magnitude of Vector:
In contrast, the scalar has only magnitude, and the vectors have both magnitude and direction. To determine the magnitude of a vector, we must first find the length of the vector. The magnitude of a vector formula denoted as 'v', is used to compute the length of a given vector ‘v’. So, in essence, this variable is the distance between the vector's initial point and to the endpoint.