Dot Product and Scalar Product of Vectors, Physics

Transcript

[Music] 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 for example if I have two vectors A and B and If I multiply these two vectors and it's in 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 AB and c and let if 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 plus sign now I will multiply the Y component of P Vector with the Y component of Q vector and that is b y 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 or dot product or 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 2 I + 3 J + 4 K and if I have another Vector B which is equal to 7 I + 5 J + K now if someone is again asking me find the dot product of a vector and B vector 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 do product of a is equal to here again I will multiply the X component of vector a with the X component of vector B and that is 2 * by 7 plus similarly I will multiply 3 * 5 + 4 * 1 and I I am getting the dot product of a vector and B Vector is equal to 2 * by 7 and I am getting 14 + 3 * 5 and I'm getting 15 + 4 * 1 and I'm getting 4 and this is equal to 33 and now remember as I told you that the dot product of two 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 ab 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'm 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 = to 2 i - 2 J + K and B is equal to 12 I + 4 J - 3 K now here I I I I try to find the angle between Vector A and B now the component of vector a a is 2 - 2 1 and the component of vector B is 12 4 and - 3 now I will use the the theorem cos Theta is equal to the dotproduct 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 f 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 and the first equation now the dot product of ab is equal to 2 * by 12 + - 2 * 4 + 1 * -3 and I'm getting 24 - 8 - 3 and I'm getting 133 now I will find the magnitude of vector and that is equal to 2 2 + - 2 2 + 1 s and I'm getting under < TK 9 and under root 9 is equal to 3 now I will find the magnitude of vector B and that is equal to under < TK 12 s + 4 2 + -3 whole s and I'm getting under < TK 1 69 and that is equal to 13 now I will again write the equ equation one cos Theta is equal to do 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'm getting 1 / 3 now Theta is equal to cos inverse 1 / 3 and it is approximately equal to 70° remember that Theta of dot product also tell us about the nature of two vectors if Theta is equal to 90° then these two vectors are perpendicular or orthagonal to each other and remember if Theta is equal to Zer then these two vectors are parallel to each other if Theta is equal to 180° 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 r we say the do product of two vectors is commutative for example if I have a vector v which is equal to 2 I + 3 J and if I have a vector U which is equal to 5 I + 7 J and now the dot product of v and U is equal to 2 * 5 + 3 * 7 and the do product of v and U is equal to 10 + 21 and I'm getting 31 now the dot product of u and v is equal to 5 * 2 + 7 * 3 and I'm getting 10 + 21 and the do product of u and v is equal to 31 so we 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 zero then this means either U Vector is equal to Z or V Vector is equal to Z are we say that the 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 one now the angle between two parall Vector is zero now what I will do I will put the value of theta in equation one and now the equation well the COS 0 remember cos 0 is equal to 1 so I will get a do 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 ab and the dot product of ab is equal to AB cos Theta now the angle between two anti parallel Vector is equal to 180° and cos 180° is equal to minus1 so the equation number one 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 [Music] vectors a