Question:
Find the direction cosines of the vector \(\hat{i}+2\hat{j}+3\hat{k}.\)
Find the direction cosines of the vector \(\hat{i}+2\hat{j}+3\hat{k}.\)
Updated On: Sep 19, 2023
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Solution and Explanation
The correct answer is:\((\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}).\)
Let \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}.\).
\(∴|\vec{a}|=\sqrt{1^2+2^2+3^2}=\sqrt{1+4+9}=\sqrt{14}\)
Hence,the direction cosines of \(\vec{a}\) are \((\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}).\)
Let \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}.\).
\(∴|\vec{a}|=\sqrt{1^2+2^2+3^2}=\sqrt{1+4+9}=\sqrt{14}\)
Hence,the direction cosines of \(\vec{a}\) are \((\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}).\)
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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.