Understanding the Vector Identities: a·b a·c and b c

The question of whether vector b c is the same as a·b a·c is a fundamental one in linear algebra and vector calculus. The answer depends on the specific conditions and interpretations of the vectors involved.

Overview of Vector Identities

Let's consider the properties of the dot product and vector magnitudes. The dot product of two vectors a and b is defined as a·b |a||b|cosθ, where |a| and |b| are the magnitudes of vectors a and b, and θ is the angle between them.

When a·b a·c

The equation a·b a·c implies that the dot product of vector a with vector b is equal to the dot product of vector a with vector c. This can happen under various conditions. Let's explore a few scenarios:

Example 1: Varying Angles but Equal Result

Consider the situation where α is the angle between vectors a and b, and β is the angle between vectors a and c. Then we have:

a·b |a||b|cosα
a·c |a||c|cosβ

For a·b a·c, it must be that |b|cosα |c|cosβ. This implies that while the magnitudes of b and c can be different, their directions relative to a can adjust to yield the same result. Mathematically, this can be achieved if the angles α and β are such that the products of their magnitudes and cosines are the same.

Example 2: Equal Magnitude but Opposite Directions

Consider the situation where b c but α β 360°. In this case, the angle β between a and c can be expressed as β 360° - α. Substituting this into the dot product equations, we get:

a·c |a||c|cos(360° - α) |a||c|cosα a·b

This shows that even if b and c are not equal, their dot products with a can be the same, provided the angles are appropriately adjusted.

When Does b c?

The converse statement, that if a·b a·c, then b c, is not always true. There are cases where b and c are not equal but their dot products with a are the same. For instance:

Both vectors could have the same magnitude but different directions relative to a. The cosine of their respective angles with a could be equal, leading to the same dot product.

Vector b - c could be perpendicular to a. That is, if a·(b - c) 0, then the dot product of a with b and c would still be the same.

In summary, while b c guarantees that a·b a·c, the reverse is not necessarily true.

Special Basis Vectors

To further illustrate, consider the basis vectors i, j, k.

For example:

i·j 0
I·k 0
j ≠ k

Here, the dot product of i with j and k is zero, but j is not equal to k. This shows that zero dot product with a vector can occur even if the vectors are not equal.

Conclusion

Understanding the relationship between vector equality and dot products requires careful consideration of the vectors' directions and magnitudes. It is important to distinguish between the conditions under which b c implies a·b a·c and the cases where the reverse is not true.

Hopefully, this explanation clarifies the nuances of vector identities in linear algebra and vector calculus. If you have any further questions, feel free to ask!