![]() ![]() One Last thing you want to note! What happens when you cross product two vectors and you get the 0 vector, what does that mean, specifically what does that tell you about the two input vectors? If θ is 0, the area will be 0! If θ is 90, the area will be at its largest! You won’t need to memorize this but it is a cool thing to note! the area will shrink while if they are perpendicular you will have the largest area that a parallelogram can be! That means the sin of the equation controls the area’s scalability. Notice how if the angle between vector A and vector B decreases. ∥a∥ ∥b∥sinθ is another way to calculate the area of the parallelogram. The area of that is ||A X B|| or the magnitude of A cross B. So notice how vectors A and B can create a parallelogram. Not many people know this one but it actually came up in an interview and I nailed it so I am writing about it here. The magnitude of the cross product tells us the area of the parallelogram spanned by the two vectors of the cross product. The resultant vector is the normal vector to the triangle! Then we take the cross product between the vectors AB and AC. Remember to make vector AB we subtract B – A. We first create a vector AB and vector AC from simple vector rules. This is one of the primary use cases of the cross product. So why is that important? Well for one, this is used to figure out the normal of a triangle! If you imagine a triangle being defined with 3 points respectively labeled as ABC, we can figure out the perpendicular vector of that triangle. The cross product returns a perpendicular vector to the two input vectors. Now if in the right-hand rule, the a and b vector were to swap, how will your hand curl? If you understood this correctly, then we would see that the thumb is pointing down and we are curling the hand upside down from the A vector to the B vector. The thumb in this case will also show you the perpendicular vector. Imagine curling your hand from the direction of the A vector towards the B vector. Your thumb shows you the perpendicular vector and its direction! Your index and middle finger show your two input vectors A and B respectively. But our big boy mathematicians have got you covered. ![]() There isn’t something in doing the cross product that tells you the direction of the perpendicular vector! I know right. Let’s quickly talk about the direction of the perpendicular vector. The key thing here is that U is on the left of V and left of W! The order of the vectors of the cross product matters! If you don’t remember these two rules, just remember the order of the cross product matters! Please! The third rule is just showing us the distribution property! Notice that U is being distributed to both v and w. One that goes in one direction and the other that goes in the opposite! This makes sense since there are two possible vectors that can be perpendicular two the input vectors of U and V. V X U is also a perpendicular vector but in the opposite direction of V X U. The first rule tells us something important. Try to remember them.įirst of all, the order of the vectors of the cross product totally matters! U X V does not equal V X U! That is the simplicity of it. ![]()
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