Compute the magnitude of the following vectors:\(\overrightarrow{a}\)=\(\hat{i}\)+\(\hat j+\hat k\);\(\overrightarrow{b}\)=2\(\hat{i}\)-7\(\hat{j}\)-3\(\hat{k}\); \(\overrightarrow{c}\)= \(\frac{1}{\sqrt 3}\hat i+\frac{1}{\sqrt 3}\hat j-\frac{1}{\sqrt 3}\hat k\)
Compute the magnitude of the following vectors:\(\overrightarrow{a}\)=\(\hat{i}\)+\(\hat j+\hat k\);\(\overrightarrow{b}\)=2\(\hat{i}\)-7\(\hat{j}\)-3\(\hat{k}\); \(\overrightarrow{c}\)= \(\frac{1}{\sqrt 3}\hat i+\frac{1}{\sqrt 3}\hat j-\frac{1}{\sqrt 3}\hat k\)
Solution and Explanation
The given vectors are: :\(\overrightarrow{a}\)=\(\hat{i}\)+\(\hat j+\hat k\);\(\overrightarrow{b}\)=2\(\hat{i}\)-7\(\hat{j}\)-3\(\hat{k}\); \(\overrightarrow{c}\)= \(\frac{1}{\sqrt 3}\hat i+\frac{1}{\sqrt 3}\hat j-\frac{1}{\sqrt 3}\hat k\)
|\(\overrightarrow{a}\)| =\(\sqrt {(1)^2+(1)^2+(1)^2}\)=\(√3 \)
|\(\overrightarrow{b}\)| =\(\sqrt{(2)^2+(-7)^2+(-3)^2} = \sqrt{4+49+9}\)
=\(√62 \)
|\(\overrightarrow{c}\)| = \(\sqrt{(\frac {1}{√3})^2+(\frac{1}{√3})^3+(\frac {-1}{√3}})^2\)
= \(\sqrt{(\frac {1}{√3})+(\frac{1}{√3})+(\frac {-1}{√3}})\) =1
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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.