Scalar product formula with an example

Scalar multiplication of two vectors yields a scalar product. Scalar multiplication is also known as the dot product. In the next section, we will see how the scalar product formula or equation is written.

Scalar product formula | equation of dot product

The scalar product of 2 vectors A and B is expressed by the following equation:
A.B = AB cos φ, where φ is the angle between the vectors, A is the magnitude of vector A and B is the magnitude of vector B.

The scalar product is also called the dot product because of the dot notation that indicates it.

scalar product or dot product equation
—— (1)
how scalar product is done
Fig 2

In figure (b), vector A is projected along the direction of vector B, as a result, a projected component of vector A is generated along the direction of vector B. And this component is A cos φ.

Similarly, in figure (c), the vector B is projected along the direction of vector A, as a result, a projected component of vector B is generated along the direction of vector A. And this component is B cos φ.

In both cases, you can see how the cos φ is generated as we are working to find out a scalar product of 2 vectors A and B.

Numerical problem solving using the scalar product or dot product

scalar product numerical question - diagram
Vector A
Vector F

Two vectors A and F are shown in the above 2 diagrams. Find the scalar or dot product of A and F.

Solution

From Figure, the magnitudes of vectors A and F are A = 10.0 and F = 20.0.

Angle θ , between them, is the difference: θ = φ − α = 110° − 35° = 75° .

Substituting these values into Equation of scalar product gives the scalar product.

A straightforward calculation gives us A.F = AF cos θ = (10.0)(20.0) cos 75° = 51.76.

Work is scalar or vector?

Work is a scalar. It’s a scalar product of 2 vectors, force, and displacement. As work is scalar, only the Magnitude of work counts.

Energy is scalar or vector?

Energy is a scalar. only the Magnitude of energy counts.

Power is scalar or vector?

Power is a scalar quantity. It has no direction attached. Magnitude of power only matters.

Scalar product formula with an example
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