Show that the vector \(\hat{i}+\hat{j}+\hat{k}\) is equally inclined to the axes OX,OY,and OZ.
Solution and Explanation
Then,
\(|\vec{a}|=\sqrt{1^2+1^2+1^2}=\sqrt{3}\)
Therefore,the direction cosines of \(\vec{a}\) are\((\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}).\)
Now,let \(α,β\),and \(γ\) be the angles formed by \(\vec{a}\) with the positive directions \(x,y\),and \(z\) axes.
Then,we have \(cosα=\frac{1}{\sqrt{3}},cosβ=\frac{1}{\sqrt{3}},cosγ=\frac{1}{\sqrt{3}}.\)
Hence,the given vector is equally inclined to axes OX,OY,and OZ.
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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.