Hello, I will explain the concept of what is dot product, how you find angle between two vectors, the properties of dot product and the application of dot product. Now what is dot product? As we know that we can add two vectors, we can subtract two vectors. Now it is time that we will multiply two vectors. For example, if I have two vectors a and b and if I multiply these two vectors and instead of multiplication sign I put the dot sign between a and b.
Now this statement will be read as a dot product of vector a and b. Now how we multiply two vectors? For example, if I have a vector p and its components is a, b and c and let f i have another vector q and its component is x, y and z. Now if someone is asking me find the dot product of p and q.
Now what I will do? I will write the dot product of p and q is equal to I will multiply the x component of p with the x component of q and that is a x and then I will put the plus sign. Now I will multiply the y component of p vector with the y component of q vector and that is by and then again I will put the plus sign and now in the last I will multiply the z component of p with the z component of q and I'm getting cz.
Now this is the multiplication of dot multiplication a dot product scalar product of P vector and Q vector. Remember that the dot product of two vectors P and Q will always give me a number. Now for example if I have a vector a which is equal to 2i plus 3j plus 4k and if I have another vector b which is equal to 7i plus 5j plus k. Now if someone is again asking me find the dot product of a Victor and B Victor now here first of all I will take the component of a and that is 2 3 and 4 and the component of B is 7 5 and 1 now the dot product of a B is equal to here again I will multiply the X component of Victor a with the X component of Victor B and that is 2 x 7 Plus similarly I will multiply 3 x 5 plus 4 multiplied by 1 and I am getting the dot product of a vector and b vector is equal to 2 multiplied by 7 and I am getting 14 plus 3 multiplied by 5 and I am getting 15 plus 4 multiplied by 1 and I am getting 4 and this is equal to 33 and now remember as I told you that the dot product of 2 vector is giving me a number now 33 is a number Now how we find angle between two vectors?
To find an angle between two vectors we use a theorem of dot product. Let the dot product of a b is equal to magnitude of a vector to the product of magnitude of b vector cos theta. Or we can rearrange this equation and I am getting cos theta is equal to a vector to the dot product of b vector divided by the magnitude of a vector to the product of magnitude of b vector. Now remember theta is the angle between vector a and b.
Now if I have a question find angle between the vectors a is equal to 2i minus 2j plus k and b is equal to 12i plus 4j minus 3k. Now here I I tried to find the angle between vector A and B. Now the component of vector a is 2, minus 2, 1 and the component of vector b is 12, 4 and minus 3. Now I will use the theorem cos theta is equal to the dot product of ab divided by the magnitude of a vector to the product of magnitude of b vector and remember first of all here I will find the dot product of ab then I will find the magnitude of vector a and then I will find the magnitude of vector b and after finding these three things I will put these three things in this equation in the first equation.
Now the dot product of a b is equal to 2 multiplied by 12 plus minus 2 multiplied by 4 plus 1 multiplied by minus 3 and I'm getting 24 minus 8 minus 3 and I'm getting 13. Now I will find the magnitude of vector a and that is equal to 2 square plus minus 2 whole square plus 1 square and I am getting under root 9 and under root 9 is equal to 3. Now I will find the magnitude of vector b and that is equal to under root 12 square plus 4 square plus minus 3 whole square and I am getting under root 169 and that is equal to 13. Now I will again write the equation 1. Cos theta is equal to dot product of AB divided by magnitude of A and magnitude of B. And cos theta is equal to I know the dot product of AB is equal to 13 divided by. And the magnitude of vector A is 3. And the magnitude of B is 13. And 13, 13 will be cancelled out and I am getting 1 divided by 3. Now theta is equal to cos inverse 1 divided by 3 and it is approximately equal to 70 degree.
Remember that theta of dot product also tell us about the nature of two vectors. If theta is equal to 90 degree then these two vectors are perpendicular or orthogonal to each other. And remember if theta is equal to 0 then these two vectors are parallel to each other. If theta is equal to 180 degree then we say these two vectors are anti-parallel to each other. Now let me tell you about the properties of dot products of vectors.
The dot product of V and U is equal to the dot product of U and V. Or we say the dot product of two vectors is commutative. For example if I have a vector V which is equal to 2i plus 3j. and if I have a vector u which is equal to phi i plus 7j and now the dot product of v and u is equal to 2 multiplied by 5 plus 3 multiplied by 7 and the dot product of v and u is equal to 10 plus 21 and I'm getting 31. Now the dot product of u and v is equal to 5 multiplied by 2 plus 7 multiplied by 3 and I'm getting 10 plus 21 and the dot product of u and v is equal to 31. So we can see that the dot product of v and u is equal to 31 and the dot product of u and v is also equal to 31. Hence we say the dot product of two vectors is commutative.
Now remember the second property of dot product is if I have two vectors the dot product of two vectors u and v is equal to 0. Then this means either u vector is equal to 0 or v vector is equal to 0 or we say that u and v vectors are perpendicular to each other. Dot product of two parallel vectors is equal to the product of its magnitude. For example, if I have two vectors a and b and I know the dot product of a and b is equal to magnitude of a vector to the product of magnitude of b vector and cos theta.
Let this is equation number 1. Now the angle between two parallel vector is 0. Now what I will do, I will put the value of theta in equation 1 and now the equation well the cos 0 remember cos 0 is equal to 1. So I will get a dot product of b is equal to magnitude of a vector to the product of magnitude of b vector. Now what about the dot product of two anti-parallel vectors. Now here again let I will consider two vectors and the dot product of two vectors let these vectors are a b and the dot product of a b is equal to a b cos theta.
Now the angle between two anti-parallel vector is equal to 180 degree. and cos 180 degree is equal to minus 1. So the equation number 1 will be shifted to the dot product of A and B is equal to minus magnitude of A vector to the product of magnitude of B vectors.