Projection of Vector a onto Vector b: A Comprehensive Guide

Understanding vector projections is crucial in many areas of mathematics and physics. This guide provides a step-by-step explanation of how to find the projection of vector onto vector , given the dot product and the vector .

Conceptual Overview and Formulae

The projection of vector onto vector can be found using the following formula:

text{proj}_{mathbf{b}} mathbf{a} frac{mathbf{a} cdot mathbf{b}}{mathbf{b} cdot mathbf{b}} mathbf{b}

This formula involves several components, namely the dot product , the dot product of , and the resultant vector scaled by the value of .

Calculating the Projection

Given the values and , we first need to calculate :

mathbf{b} cdot mathbf{b} 2^2 6^2 3^2 4 36 9 49

With and , we can substitute into our projection formula:

text{proj}_{mathbf{b}} mathbf{a} frac{8}{49} mathbf{b}

Next, we substitute into the formula:

text{proj}_{mathbf{b}} mathbf{a} frac{8}{49} (2mathbf{i} 6mathbf{j} 3mathbf{k})

Finally, we distribute frac{8}{49} across mathbf{b} to find the projection vector:

text{proj}_{mathbf{b}} mathbf{a} left(frac{16}{49}right)mathbf{i} left(frac{48}{49}right)mathbf{j} left(frac{24}{49}right)mathbf{k}

Therefore, the projection of vector onto vector is:

text{proj}_{mathbf{b}} mathbf{a} frac{16}{49}mathbf{i} frac{48}{49}mathbf{j} frac{24}{49}mathbf{k}

Understanding the Geometric Interpretation

The projection vector represents the component of vector that is in the direction of vector . Geometrically, this projection can be interpreted as the length of the vector if it were directly projecting onto .

Additionally, the magnitude of the projection vector gives the scalar projection of vector onto vector , which is given by:

frac{|mathbf{a} cdot mathbf{b}|}{|mathbf{b}|} frac{8}{7}

This is because the magnitude of is |mathbf{b}| 7sqrt{1} 7.

Additional Key Concepts

1. Unit Vector: A unit vector in the direction of vector can be obtained by normalizing .

mathbf{b}_{text{unit}} frac{mathbf{b}}{|mathbf{b}|} frac{2mathbf{i} 6mathbf{j} 3mathbf{k}}{7}

2. Dot Product: The dot product provides a scalar result that can be used to find the projection.

3. Orthogonal Projection: The orthogonal projection of vector onto vector can be found using the formula:

text{proj}_{mathbf{b}} mathbf{a} frac{mathbf{a} cdot mathbf{b}}{mathbf{b} cdot mathbf{b}} mathbf{b}

Conclusion

In summary, finding the projection of vector onto vector involves several steps, including calculating the dot product, the magnitude of , and substituting these values into the projection formula. The result is a vector that represents the component of in the direction of .

Understanding these concepts is essential for advanced studies in vector algebra and its applications in fields such as physics, engineering, and computer science.