Can someone explain why (𝑎 • 𝑏) × 𝑐 exist if a, b, and c are vectors?
(𝑎 • 𝑏) × 𝑐
(𝑎 • 𝑏) is a scalar, if a and b are vectors
then (𝑎 • 𝑏) × 𝑐 would simply a scalar multiple of the vector c
Your question is incorrectly worded.
the cross product is only defined between two vectors.
You cannot take the cross-product of a scalar and a vector.
correct my bad
Certainly! In order to understand why the expression (𝑎 • 𝑏) × 𝑐 exists when 𝑎, 𝑏, and 𝑐 are vectors, let's break it down step by step.
First, let's look at the 𝑎 • 𝑏 part of the expression. The dot product (•) between two vectors is a mathematical operation that yields a scalar (a single value). The dot product is calculated by multiplying the corresponding components of the vectors and then adding them together. In other words, if 𝑎 = [𝑎₁, 𝑎₂, 𝑎₃] and 𝑏 = [𝑏₁, 𝑏₂, 𝑏₃], then 𝑎 • 𝑏 = 𝑎₁𝑏₁ + 𝑎₂𝑏₂ + 𝑎₃𝑏₃.
Next, let's consider the × 𝑐 part of the expression. The cross product (×) between two vectors is a mathematical operation that yields a vector. The cross product is calculated using a specific formula involving the components of the two vectors. If 𝑟 is the resulting vector from the cross product of 𝑎 and 𝑏, then 𝑟 = [𝑏₂𝑎₃ - 𝑏₃𝑎₂, 𝑏₃𝑎₁ - 𝑏₁𝑎₃, 𝑏₁𝑎₂ - 𝑏₂𝑎₁].
So, now let's put it all together. When we have the expression (𝑎 • 𝑏) × 𝑐, it means that we first calculate the dot product of 𝑎 and 𝑏 (which gives us a scalar), and then we take the cross product of the resulting scalar and 𝑐 (which gives us a vector).
In summary, (𝑎 • 𝑏) × 𝑐 exists because the dot product yields a scalar and the cross product operates between a scalar and a vector, resulting in the desired vector.