3D Dot Product Calculator

What is a 3D Dot Product?

Imagine you have two magic arrows in a big room. The 3D dot product is like a special number that tells us how much these arrows agree with each other. If they point in similar directions, the dot product is a big positive number. If they point in opposite directions, it's a big negative number. And if they're at right angles, it's zero!

How to Calculate the 3D Dot Product

To find the dot product of our magic arrows (3D vectors), we use a simple math trick. We multiply the matching parts of our arrows and then add these numbers together. It's like asking our arrows how much they agree in each direction and then combining all that agreement into one number!

Formula

If we have two 3D vectors \(\vec{a} = (a_x, a_y, a_z)\) and \(\vec{b} = (b_x, b_y, b_z)\), their dot product is:

\[ \vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z \]

Where:

• \(a_x\) and \(b_x\) are how far the arrows point in the left-right direction
• \(a_y\) and \(b_y\) are how far the arrows point in the up-down direction
• \(a_z\) and \(b_z\) are how far the arrows point in the forward-backward direction

Calculation Steps

1. Multiply the x parts: \(a_x \times b_x\)
2. Multiply the y parts: \(a_y \times b_y\)
3. Multiply the z parts: \(a_z \times b_z\)
4. Add all these products together

Example

Let's find the dot product of two magic arrows: \(\vec{a} = (1, 2, 3)\) and \(\vec{b} = (4, 5, 6)\)

1. Multiply x parts: \(1 \times 4 = 4\)
2. Multiply y parts: \(2 \times 5 = 10\)
3. Multiply z parts: \(3 \times 6 = 18\)
4. Add them all: \(4 + 10 + 18 = 32\)

So, the dot product of our magic arrows \(\vec{a}\) and \(\vec{b}\) is 32!

Visual Representation