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For what value of $\alpha $ are the vectors $a = \{ 2,3, - 4\} ,\,and\,b = \{ \alpha , - 6,8\} $ are parallel.

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Last updated date: 16th Jul 2024
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Answer
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Hint: In mathematics a vector is an object which has direction and magnitude. Magnitude defines the size of the vector. It is represented by a line with an arrow. Where the length of the line is magnitude and arrows show the direction of the vector. The projection of a vector on a plane is an orthogonal projection. And rejection of a vector from a plane is its orthogonal projection in a straight line.
We should know that the parallel vector's ratio of coefficients will equal. Using this concept for a given question we can find the unknown variable.

Complete step by step solution:
Given,
Vector a, $\vec a = \{ 2,3 - 4\} $
Vector b, $\vec b = \{ \alpha , - 6,8\} $
The given vector $a$ and $b$ are parallel
So the value of $\alpha $ a will be find by
$ \Rightarrow \dfrac{2}{\alpha } = \dfrac{3}{{ - 6}} = \dfrac{{ - 4}}{8}$
$ \Rightarrow \dfrac{2}{\alpha } = \dfrac{1}{{ - 2}} = \dfrac{{ - 1}}{2}$
So here
$ \Rightarrow \dfrac{2}{\alpha } = - \dfrac{1}{2}$
Simplify
$ \Rightarrow - 1 \times \alpha = 2 \times 2$
$ \Rightarrow \alpha = - 4$
The value of $\alpha = - 4$ where vectors are parallel.
So, the correct answer is “$\alpha = - 4$”.

Note: Scalar quantities are those quantities which are described by their magnitude. They don’t have any direction. Some examples of scalar quantity are- volume, mass, density, speed. The two scalar quantities can also be multiplied or divided to form the derived scalar quantity. Historically vectors were introduced in geometry and physics before the formalization of the concept of vector space.
Vectors have many real-life applications, including situations involving force or velocity. For example, consider the forces acting on a boat crossing a river. The boat's motor generates a force in one direction, and the current of the river generates a force in another direction. Both forces are vectors