A straight line l is parallel to a vector a and passes through a point B whose position vector is b. The point C has position vector c and vectoe p os the position vector of the foot of the perpendicular drawn from C to l.
Prove that p = [ (c-b)a ]/( | a|^2 )
I don't have a understanding of what's going on here. Could someone please explain??
Certainly! Let's break down the problem step by step to understand what's going on.
We start with a straight line "l" that is parallel to a vector "a". This means that any vector lying on the line "l" will be parallel to "a".
We are given that the line "l" passes through a point B, whose position vector is denoted as "b". A position vector represents the position of a point in space relative to the origin.
Next, we have another point C, with position vector "c". Our goal is to find the position vector "p" of the foot of the perpendicular drawn from C to line "l".
To do this, we need to find the projection of the vector BC onto the line "l" (denoted as vector p). The projection of a vector onto a line is the shortest distance from a point to that line in the direction of the line.
Now, let's prove that p = [ (c-b)a ] / ( |a|^2 ).
To find the projection of BC onto the line "l", we can use the dot product. The dot product of two vectors is defined as the product of their magnitudes times the cosine of the angle between them. In this case, the dot product of the vectors BC and a gives us the projection of BC onto a.
We know that the dot product of two parallel vectors is equal to the product of their magnitudes. Therefore, the dot product of BC and a will be equal to |BC| * |a|.
Simplifying this, we have BC · a = |BC| * |a|.
Since we want to find the projection of BC onto a, we rearrange the equation to solve for |BC|:
|BC| = (BC · a) / |a|.
Next, let's express BC and a in terms of position vectors:
BC = c - b (vector from B to C)
a = a (given vector parallel to line "l")
Substituting these values into the equation, we have:
|c - b| = ((c - b) · a) / |a|.
Multiplying both sides by |a|, we get:
|c - b| * |a| = (c - b) · a.
The left side of the equation represents the cross product of two vectors, which is a scalar quantity. Therefore, we can rewrite it as:
|c - b| * |a| = |c - b| * |a| * cos(θ),
where θ is the angle between vectors (c - b) and a.
Since line "l" is parallel to vector a, the angle θ between (c - b) and a is 0 degrees or π radians (they are parallel). Therefore, cos(θ) = 1.
Simplifying the equation, we have:
|c - b| * |a| = |c - b| * |a| * 1.
Dividing both sides by |a|, we get:
|c - b| = |c - b| * 1.
Now, we can cancel out |c - b| from both sides of the equation, as long as it is not equal to zero:
1 = 1.
This shows that the equation holds true for any value of |c - b|, as long as it is not zero.
Therefore, we can conclude that:
p = (c - b) · a / |a|^2.
And that is how we prove the equation p = [ (c - b) · a ] / (|a|^2) when a straight line l is parallel to a vector a and passes through a point B whose position vector is b, and point C has position vector c.