Write two different vectors having the same direction.
Write two different vectors having the same direction.
Solution and Explanation
Consider \(\overrightarrow p=(\hat i+\hat j+\hat k)\) and \(\overrightarrow q=(2\hat i+2\hat j+2\hat k)\).
The direction cosines of \(\overrightarrow p\) are given by,
\(I=\frac{1}{\sqrt{1^2+1^2+1^2}}=\frac{1}{\sqrt 3}\), \(m=\frac{1}{\sqrt{1^2+1^2+1^2}}=\frac{1}{\sqrt 3}\), and \(n=\frac{1}{\sqrt{1^2+1^2+1^2}}=\frac{1}{\sqrt 3}\).
The direction cosines of \(\overrightarrow q\) are given by
\(I=\frac{2}{\sqrt{2^2+2^2+2^2}}=\frac{2}{2\sqrt3}\frac{1}{\sqrt 3}\), \(m=\frac{2}{\sqrt{2^2+2^2+2^2}}=\frac{2}{2\sqrt3}\frac{1}{\sqrt 3}\), and \(n=\frac{2}{\sqrt{2^2+2^2+2^2}}=\frac{2}{2\sqrt3}\frac{1}{\sqrt 3}\).
The direction cosines of \(\overrightarrow p\) and \(\overrightarrow q\) are the same. Hence, the two vectors have the same direction.
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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:
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