Vector Calculator

I found this great website that helps with doing vector operations. It can compute magnitudes, dot products, cross products, and more for any 2D or 3D vectors. Also, it can show an explanation to how it reached each answer, so you can follow along if you are confused. I hope this helps you with your future homework or studying!
Click here for the link to this site.

Vectors in Space

Earlier this week, we learned about vectors in space. A vector is given by:

v = <v₁, v₂, v₃> (component form)
or
v = v₁i + v₂j + v₃k (unit vector form)

Many operations can be performed with vectors, and, to demonstrate, I will perform these operations with the following vectors:

u = <2, 8, 4>
v = <6, 12, 10>

To find the length (also known as magnitude) of the vector:

||u|| = √(u₁² + u₂² + u₃²)
||u|| = √(2² + 8² + 4²) = √(4 + 64 + 16) = √84 = ~9.165

To find the sum of the two vectors:

u + v =  <u₁ + v₁, u₂ + v₂, u₃ + v₃>
u + v = <2 + 6, 8 + 12, 4 + 10> = <8, 20, 14>

To perform a scalar multiplication on the vector:

cu = <cu₁, cu₂, cu₃>
4u = <4(2), 4(8), 4(4)> = <8, 32, 16>

To find the dot product of the vectors:

u • v = u₁v₁ + u₂v₂ + u₃v₃
u • v = 2(6) + 8(12) + 4(10) = 12 + 96 + 40 = 148

For more information on vector operations, you can visit section 11.2 on page 837 of our book.

The Cross Product of Two Vectors

The cross product of two vectors is a vector that is perpendicular to both. The easiest way to find it is to use matrices, but you can also use this formula:

To find the cross product by using a matrix, create a 3x3 matrix. The first row should be the variables i, j, and k. The second row should be the values of the first vector, and the third row should be the values of the second vector.

For example, this is what the matrix would look like when finding the cross product of the vectors <1, 2, 1> and <3, 1, 2>:

i j k

1 2 1

3 1 2

Now, find the discriminant of this matrix:

Finally, we see that the cross product of these two vectors is <3, 1, -5>.

5 3D Graphs