# Vector question (scalar products)

• jaejoon89
In summary, to find scalars s and t such that C - sA - tB is perpendicular to both A and B, one can either set Q = C - sA - tB and solve a system of 3 equations in 3 variables using the fact that Q is a scalar multiple of (3,3,-3), or form a system of 2 equations in 2 variables by setting Q.A = Q.B = 0.
jaejoon89
Find scalars s, t s.t. C - sA - tB is perpendicular to A and B

A = i + j + 2k
B = 2i - j + k
C = 2i - j + 4k

I took the cross product of A and B
(1, 1, 2) x (2, -1, 1) =
|i j k|
|1 1 2|
|2 -1 1|
= 3i + 3j - 3k

OK, that should be perpendicular to both vectors A and B I'm guessing...
But how do I still determine the scalars s and t -- there is only one equation!

Ill let Q = C - sA - tB.

If Q is perpendicular to both A and B, and the vector (3,3,-3) is also perpendicular to both A and B (as you have calculated), then it should follow that Q is a scalar multiple of (3,3,-3) right?

i.e.
k(3,3,-3) = Q

Now you have a system of 3 equations in 3 variables (s,t,k).

____

Another way that you could solve it would be to realize that you need to find s,t such that Q.A = Q.B = 0 (a zero scalar product implies orthogonality), and form a system of two equations in two variables (s,t).

## 1. What is a scalar product?

A scalar product, also known as a dot product, is a mathematical operation that takes two vectors and produces a scalar quantity. It is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them.

## 2. How is a scalar product calculated?

To calculate a scalar product, you first need to determine the magnitude of the two vectors and the angle between them. Then, multiply the magnitudes and the cosine of the angle. The result will be a scalar quantity.

## 3. What is the significance of the scalar product?

The scalar product is significant because it helps in determining if two vectors are parallel or perpendicular to each other. It also helps in calculating the work done by a force in a particular direction and the projection of one vector onto another.

## 4. Can a scalar product be negative?

Yes, a scalar product can be negative. It depends on the angle between the two vectors. If the angle is greater than 90 degrees, the scalar product will be negative. If the angle is less than 90 degrees, the scalar product will be positive.

## 5. In what real-life situations is the scalar product used?

The scalar product is commonly used in physics and engineering, such as calculating work done by a force, finding the angle between two forces, and determining the components of a force in a particular direction. It is also used in computer graphics to determine the lighting and shading of 3D objects.

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