Question

What formula for the dot product makes these geometric properties clear? Explain.

two explainations please

Answer 1

Solution : - The dot product is fundamentally a projection. As shown in Figure 1, the dot product of a vector with a unit vector is the projection of that vector in the direction given by the unit vector. This leads to the geometric formula v * w = |v||w | cos θ for the dot product of any two vectors v and w .

One can see immediately from the formula that the dot product a⋅ba⋅b is positive for acute angles and negative for obtuse angles.

The formula demonstrates that the dot product grows linearly with the length of both vectors and is commutative, i.e., a⋅b=b⋅a

2) The dot product between two vectors is based on the projection of one vector onto another.

Let's imagine we have two vectors a and b, and we want to calculate how much of a is pointing in the same direction as the vector b. We want a quantity that would be positive if the two vectors are pointing in similar directions, zero if they are perpendicular, and negative if the two vectors are pointing in nearly opposite directions.