Write two different vectors having same magnitude.
Solution and Explanation
Consider \(\vec a\) \(= (\hat i-2\hat j+3\hat k)\) and \(\vec b\) =\( (2\hat i+\hat j-3\hat k)\).
It can be observed that:
\(|\vec a|\)= \(\sqrt {1^2+(-2)^2+3^2}\) = \(\sqrt {1+4+9}\) = \(\sqrt {14}\)
\(|\vec b|\) = \(\sqrt {2^2+1^2+(-3)^2}\) = \(\sqrt {4+1+9}\) = \(\sqrt {14}\)
Hence, \(\vec a\) and \(\vec b\) are two different vectors having the same magnitude. The vectors are different because they have different directions.
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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.