What is Cross product of perpendicular vectors ? In this article you will learn about cross product of two perpendicular vectors. The vectors whose scalar product is zero are known as Perpendicular Vectors. So, the basic definition of a perpendicular vector is that let’s suppose that two vectors A and B are perpendicular if and only if their scalar product is Zero.
Let A̅ = (Ax, Ay) and B̅ = (Bx, By)
Vectors A̅ and B̅ are perpendicular if and only if A̅•B̅ = 0
(Ax, Ay)•(Bx , By) = Ax Bx + Ay By
Hence vectors A̅ and B̅ are perpendicular if and only if
Ax Bx + Ay By = 0
Cross Product of Perpendicular Vectors
Vectors are multiplied with each other but it is not a simple process it is a complicated one. There are two types of vector multiplication. the first one is Dot Product and the second one is Cross Product.
The Cross Product of two vectors is another perpendicular vector to the two vectors. Perpendicular Vectors are also known by the name of Orthogonal Vectors. The direction of the resultant vector can be determined by the Right Hand’s Thumb Rule. The thumb (u) and index finger (v) are held perpendicularly to one another and represent the vector and the middle finger perpendicularly to the index and thumb indicate the direction of the resultant vector.
Let’s assume that if two vectors A̅ x B̅ and B̅ x A̅ are antiparallel to each other then the vector multiplication is said to be known as Anti commutative Property. The Anti commutative property means the vector product reverses the sign when the order of multiplication is reversed:
A̅ x B̅ = – B̅ x A̅
Let’s understand with the help of Example:-
Write XYZ order twice which will be like XYZXYZ. The next step is to find the perfect pattern:
- XY => Z (X cross Y is Z)
- YZ => X (Y cross Z is X; loop around: Y to Z to X)
- ZX => Y
At this stage, XY and YX have opposite signs as they are set in the forward and backward direction of the XYZXYZ setup respectively.
So we can calculate:
X x Y = (1, 0, 0) x (0, 1, 0) = (0, 0, 1) = Z
And this whole process is because X cross Y is positive which is equivalent to Z in a right-hand coordinate system.
Example of Cross Product of Perpendicular Vectors
Question: Find the real numbers a so that the vectors A = (2a, 16) and B = (3a+2, -2) are perpendicular?
Solution: The condition for two vectors A = (Ax, Ay) and B = (Bx, By) to be perpendicular is: Ax Bx + Ay By = 0
After rewriting the above condition, we obtain the following equation:
2a(3a + 12) + 16(-3) = 0
Expanding and rearranging the above equation:
3a^2 + 2a – 16 = 0
After solving the equation we obtained that the value of a and b is:
a = 2 and b = – 8 / 3
Use cross product of two vectors calculator to calculate cross product of two vectors. If you have any doubts comment down. We will get back to you. Thank you.
Read More: Cross product calculator online