You will often see proj (Vector a onto Vector b) written as projb⃗ a⃗ .Show that projb⃗ a⃗ =(a⃗ ⋅b⃗/ b⃗ ⋅b⃗ )*b⃗ .
To prove that projb⃗ a⃗ =(a⃗ ⋅b⃗/ b⃗ ⋅b⃗ )*b⃗, we can use the geometric definition of projecting a vector onto another vector.
Let's start by defining the projection of vector a⃗ onto vector b⃗ as projb⃗ a⃗. This projection is a scalar multiple of vector b⃗, which means it lies along the same direction as b⃗.
Now, we can express the projection projb⃗ a⃗ in terms of vector b⃗. To find this, we need to determine how much of vector a⃗ lies in the same direction as vector b⃗. We can do this by finding the scalar multiple of vector b⃗ that gives us the projection.
Let's call this scalar multiple x. So, the projection of vector a⃗ onto vector b⃗ can be written as:
projb⃗ a⃗ = x * b⃗
To find the value of x, we can use the dot product. The dot product of two vectors a⃗ and b⃗ is defined as:
a⃗ ⋅ b⃗ = |a⃗ ||b⃗ |cosθ
where |a⃗ | and |b⃗ | are the magnitudes of the vectors, and θ is the angle between them.
In this case, the dot product between vector a⃗ and vector b⃗ is:
a⃗ ⋅ b⃗ = |a⃗ ||b⃗ |cosθ
Now, let's multiply both sides of the equation by vector b⃗ :
(a⃗ ⋅ b⃗) * b⃗ = (|a⃗ ||b⃗ |cosθ) * b⃗
Using the distributive property of multiplication, we can write this as:
(a⃗ ⋅ b⃗) * b⃗ = |a⃗ | * |b⃗ | * cosθ * b⃗
Since cosθ * |b⃗ | gives us the magnitude of the projection b⃗, we can simplify further:
(a⃗ ⋅ b⃗) * b⃗ = |a⃗ | * (b⃗ ⋅ b⃗ ) * b⃗
Finally, we divide both sides of the equation by (b⃗ ⋅ b⃗ ):
projb⃗ a⃗ = (a⃗ ⋅ b⃗) * b⃗ / (b⃗ ⋅ b⃗ )
Therefore, we have shown that projb⃗ a⃗ =(a⃗ ⋅b⃗/ b⃗ ⋅b⃗ )*b⃗.