You will often see proj (Vector a onto Vector b) written as proj b⃗ a⃗ .Show that projb⃗ a⃗ =(a⃗ ⋅b⃗ b⃗ ⋅b⃗ )b⃗ .
Your notation is a mystery to me.
The magnitude of the projection of A on to A is |A| cos T
where T is the angle between A and B
but |A| cos T = |A| |B| cos T /|B|
which is
A dot B /|B|
Now we need a unit vector in direction B
That is B/|B|
so in the end we need
A dot B * B /|B|^2
cxcxcx
To prove that proj b⃗ a⃗ = (a⃗ ⋅ b⃗ / (b⃗ ⋅ b⃗)) * b⃗, we need to start with the definition of the projection of a vector a⃗ onto a vector b⃗.
The projection of a vector a⃗ onto a vector b⃗ can be defined as the vector component of a⃗ in the direction of b⃗. To find this, we need to determine how much of a⃗ lies along b⃗.
Let's denote the projection of a⃗ onto b⃗ as proj b⃗ a⃗. It can be written as a scalar multiple of b⃗; let's call this scalar k. So, proj b⃗ a⃗ = k * b⃗.
To find the value of k, we need to find the component of a⃗ that lies along b⃗. This can be done using the dot product.
The dot product of two vectors a⃗ and b⃗ is defined as the magnitude of a⃗ times the magnitude of b⃗ times the cosine of the angle between them.
The dot product of a⃗ and b⃗ can be written as:
a⃗ ⋅ b⃗ = |a⃗| * |b⃗| * cos(theta)
Where |a⃗| and |b⃗| are the magnitudes of the vectors a⃗ and b⃗, respectively, and theta is the angle between them.
Now, let's consider the dot product between a⃗ and b⃗:
a⃗ ⋅ b⃗ = |a⃗| * |b⃗| * cos(theta)
Rearranging the equation, we get:
a⃗ ⋅ b⃗ = (a⃗ ⋅ b⃗) * (|b⃗| * cos(theta))
Dividing both sides by (|b⃗| * cos(theta)), we can solve for (a⃗ ⋅ b⃗):
(a⃗ ⋅ b⃗) = a⃗ ⋅ b⃗ / (|b⃗| * cos(theta))
Now, we know the dot product of a⃗ and b⃗ can also be written as:
a⃗ ⋅ b⃗ = |a⃗| * |b⃗| * cos(theta)
But |b⃗| * cos(theta) is actually the magnitude of b⃗ projected onto a unit vector in the direction of b⃗. So, we can rewrite |b⃗| * cos(theta) as |b⃗| * (b⃗ / |b⃗|).
Substituting this back into the equation, we get:
(a⃗ ⋅ b⃗) = a⃗ ⋅ (|b⃗| * (b⃗ / |b⃗|))
Simplifying further:
(a⃗ ⋅ b⃗) = |b⃗| * (a⃗ ⋅ b⃗) / |b⃗|
Now, substituting this equation into the original equation for proj b⃗ a⃗:
proj b⃗ a⃗ = k * b⃗
We can replace k with (a⃗ ⋅ b⃗) / |b⃗|:
proj b⃗ a⃗ = (a⃗ ⋅ b⃗) / |b⃗| * b⃗
Finally, since |b⃗| is the magnitude of vector b⃗, we can express it as (b⃗ ⋅ b⃗)^(1/2):
proj b⃗ a⃗ = (a⃗ ⋅ b⃗) / (b⃗ ⋅ b⃗)^(1/2) * b⃗
To simplify the expression, multiply the numerator and denominator by (b⃗ ⋅ b⃗)^(1/2):
proj b⃗ a⃗ = (a⃗ ⋅ b⃗ / (b⃗ ⋅ b⃗)) * (b⃗ ⋅ b⃗)^(1/2) * b⃗
And since (b⃗ ⋅ b⃗)^(1/2) is just the magnitude of b⃗, we can rewrite it as:
proj b⃗ a⃗ = (a⃗ ⋅ b⃗ / (b⃗ ⋅ b⃗)) * b⃗
Therefore, we have proved that proj b⃗ a⃗ = (a⃗ ⋅ b⃗ / (b⃗ ⋅ b⃗)) * b⃗.