How can you calculate the magnitude of a vector using scalar products?

To calculate the magnitude of a vector using scalar products, the correct approach involves multiplying the vector by itself to find its squared length and then taking the square root of that result. This method is based on the concept of the dot product in vector mathematics.

When you have a vector represented in component form as ( \vec{v} = (v_x, v_y) ), the scalar product of the vector with itself is computed as:

[

\vec{v} \cdot \vec{v} = v_x^2 + v_y^2

]

The magnitude ( ||\vec{v}|| ) is then derived from this expression by taking the square root:

[

||\vec{v}|| = \sqrt{v_x^2 + v_y^2}

]

This process effectively utilizes the properties of scalar products to yield the length of the vector. The result confirms that the magnitude of a vector can be calculated accurately through this relationship established by the dot product.