Question:
For given vectors,\(\vec{a}=2\hat{i}-\hat{j}+2\hat{k}\) and \(\vec{b}=-\hat{i}+\hat{j}-\hat{k}\),find the unit vector in the direction of the vector \(\vec{a}+\vec{b}.\)
For given vectors,\(\vec{a}=2\hat{i}-\hat{j}+2\hat{k}\) and \(\vec{b}=-\hat{i}+\hat{j}-\hat{k}\),find the unit vector in the direction of the vector \(\vec{a}+\vec{b}.\)
Updated On: Sep 19, 2023
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Solution and Explanation
The correct answer is:\(\frac{1}{2}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}.\)
The given vectors are \(\vec{a}=2\hat{i}-\hat{j}+2\hat{k}\) and \(\vec{b}=-\hat{i}+\hat{j}-\hat{k}\)
\(\vec{a}=2\hat{i}-\hat{j}+2\hat{k}\)
\(\vec{b}=-\hat{i}+\hat{j}-\hat{k}\)
\(∴\vec{a}+\vec{b}=(2-1)\hat{i}+(-1+1)\hat{j}+(2-1)\hat{k}=1\hat{i}+0\hat{j}+1\hat{k}=\hat{i}+\hat{k}\)
\(|\vec{a}+\vec{b}|=\sqrt{1^2+1^2}=\sqrt{2}\)
Hence,the unit vector in the direction of \((\vec{a}+\vec{b})\) is
\(\frac{(\vec{a}+\vec{b})}{|\vec{a}+\vec{b}|}=\frac{\hat{i}+\hat{k}}{\sqrt{2}}=\frac{1}{2}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}.\)
The given vectors are \(\vec{a}=2\hat{i}-\hat{j}+2\hat{k}\) and \(\vec{b}=-\hat{i}+\hat{j}-\hat{k}\)
\(\vec{a}=2\hat{i}-\hat{j}+2\hat{k}\)
\(\vec{b}=-\hat{i}+\hat{j}-\hat{k}\)
\(∴\vec{a}+\vec{b}=(2-1)\hat{i}+(-1+1)\hat{j}+(2-1)\hat{k}=1\hat{i}+0\hat{j}+1\hat{k}=\hat{i}+\hat{k}\)
\(|\vec{a}+\vec{b}|=\sqrt{1^2+1^2}=\sqrt{2}\)
Hence,the unit vector in the direction of \((\vec{a}+\vec{b})\) is
\(\frac{(\vec{a}+\vec{b})}{|\vec{a}+\vec{b}|}=\frac{\hat{i}+\hat{k}}{\sqrt{2}}=\frac{1}{2}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}.\)
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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.