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.