Question:
Show that the vectors \(2\hat{i}-3\hat{j}+4\hat{k}\) and \(-4\hat{i}+6\hat{k}-8\hat{k}\) are collinear.
Show that the vectors \(2\hat{i}-3\hat{j}+4\hat{k}\) and \(-4\hat{i}+6\hat{k}-8\hat{k}\) are collinear.
Updated On: Sep 19, 2023
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Solution and Explanation
Let \(\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}\)and \(\vec{b}=-4\hat{i}+6\hat{k}-8\hat{k}\).
It is observed that \(\vec{b}=-4\hat{i}+6\hat{k}-8\hat{k}\)\(=-2(2\hat{i}-3\hat{j}+4\hat{k})\)\(=-2\vec{a}\)
\(∴\vec{b}=λ\vec{a}\)
where,
\(λ=-2\)
Hence,the given vectors are collinear.
It is observed that \(\vec{b}=-4\hat{i}+6\hat{k}-8\hat{k}\)\(=-2(2\hat{i}-3\hat{j}+4\hat{k})\)\(=-2\vec{a}\)
\(∴\vec{b}=λ\vec{a}\)
where,
\(λ=-2\)
Hence,the given vectors are collinear.
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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.