Find a vector in a direction of vector \(5\hat{i}-\hat{j}+2\hat{k}\) which has magnitude 8units.
Solution and Explanation
Let \(\vec{a}=5\hat{i}-\hat{j}+2\hat{k}\)
\(∴|\vec{a}|=\sqrt{5^2+(-1)^2+2^2}=\sqrt{25+1+4}=\sqrt{30}\)
\(∴\hat{a}=\frac{\vec{a}}{|\vec{a}|}=\frac{5\hat{i}-\hat{j}+2\hat{k}}{\sqrt{30}}\)
Hence,the vector in the direction of vector \(5\hat{i}-\hat{j}+2\hat{k}\) which has magnitude 8units is given by,
\(8\hat{a}=\frac{8(5\hat{i}-\hat{j}+2\hat{k})}{\sqrt{30}}\)
\(=\frac{40}{\sqrt{30}}\hat{i}-\frac{8}{\sqrt{30}}\hat{j}+\frac{16}{\sqrt{30}}\hat{k}\)
\(=8(\frac{5\vec{i}-\vec{j}+2\vec{k}}{\sqrt{30}})\)
\(=\frac{40}{\sqrt{30}}\vec{i}\frac{-8}{√30}\vec{j}\frac{+16}{√30}\vec{k}\)
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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.