Dot product - Orthogonal

You may have to include s= -1. Good job!In summary, by evaluating the dot product, we can find that the values of the scalar s for which the two vectors b=X+sY and c=X-sY are orthogonal are s=1 and s=-1. This is achieved by setting the dot product of the two vectors equal to 0. A sketch of the two vectors would show them perpendicular to each other.
  • #1
By evaluating the dot product,

find the values of the scalar s for which the two vectors
b=X+sY and c=X-sY
are orthogonal
also explain your answers with a sketch:

my working

(X,sY).(X,-sY) has to equal 0 for them to be orthogonal

x.x = 1 since they are unit vectors
sY.-sY = -1 to make the whole thing 0

s = 1
1*y . -1*y = -1

1-1 =0

sketch would be two vectors perpendicular to one another?
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  • #2
could someone please inform me If my work is correct?
  • #3
(X + sY).(X - sY) = 0
==> X.X + sY.X - sX.Y -s2Y.Y = X.X - s2Y.Y = 0
==> 1 - s2 = 0

You found one solution for s; there are two.

What is the dot product of two vectors?

The dot product of two vectors is a mathematical operation that results in a scalar value. It is calculated by multiplying the corresponding components of the two vectors and adding them together.

How can the dot product be used to determine if two vectors are orthogonal?

If the dot product of two vectors is equal to zero, then the two vectors are orthogonal. This means that the two vectors are perpendicular to each other, and their angle is 90 degrees.

What is the geometric interpretation of the dot product?

The dot product can be interpreted as the product of the lengths of two vectors and the cosine of the angle between them. This means that the dot product is a measure of how much the two vectors are aligned with each other.

Can the dot product be negative?

Yes, the dot product can be negative. This indicates that the two vectors are in opposite directions, and their angle is greater than 90 degrees.

How is the dot product used in applications?

The dot product has many applications in mathematics, physics, and engineering. It can be used to calculate work done, determine the angle between two vectors, and solve problems involving forces and velocities. It is also used in computer graphics and machine learning algorithms.

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