Vector Dot Product using Matrix (2024)

FAQs

How do you dot product a vector with a matrix? ›

If we multiply xT (a 1×n matrix) with any n-dimensional vector y (viewed as an n×1 matrix), we end up with a matrix multiplication equivalent to the familiar dot product of x⋅y: xTy=[x1x2x3⋯xn][y1y2y3⋮yn]=x1y1+x2y2+x3y3+…

Dot Product of Vectors

The scalar product of two vectors a and b of magnitude |a| and |b| is given as |a||b| cos θ, where θ represents the angle between the vectors a and b taken in the direction of the vectors.

Matrix-vector product

So, if A is an m×n matrix (i.e., with n columns), then the product Ax is defined for n×1 column vectors x. If we let Ax=b, then b is an m×1 column vector. In other words, the number of rows in A (which can be anything) determines the number of rows in the product b.

Matmul requires that the number of columns in the first matrix matches the number of rows in the second. The dot product, on the other hand, can be performed on vectors of any length.

Dot products are done between the rows of the first matrix and the columns of the second matrix. Thus, the rows of the first matrix and columns of the second matrix must have the same length. I want to emphasize an important point here: The length of a row is equal to the number of columns.

The dot product is not commutative in general and the examples in the Documentation Center for Dot explicitly show this (Examples/Basic Examples/Products of matrices and vectors).

The dot product of two vectors is the sum of the products of their corresponding components. It is the product of their magnitudes multiplied by the cosine of the angle between them. A vector's dot product with itself is the square of its magnitude.

The dot product of vectors a and unit vector u is the projection of a onto u, i.e., a⋅u=∥a∥cosθ, where θ is the angle between a and u. This expression does not involve the magnitude of u since it is normalized to be length one.

Dot products are all about projections. That is, calculating how much of one vector lies in the direction of the other. For this reason, the dot product has a cosine in the formula because it should be maximum when the angle between the vectors θ is zero.

That is, the product of a matrix A (on the left) with a vector (on the right) is the linear combination of the columns of A obtained by using the entries of x as weights. We call Ax a product and use multiplicative notation for reasons that will become clear shortly.

How is vector related to matrix? ›

If a matrix has only one row or only one column it is called a vector. A matrix having only one row is called a row vector. is a row vector because it has only one row. A matrix having only one column is called a column vector.

That is, the dot product is the summation of the products of the ith element of each vector. Note that for a dot product to be defined, both vectors must have the same number of elements. multiply them together, then sum the products. So A ∙ B = 1(4) + 3(5) + 4(2) = 4 + 15 + 8 = 27.

The dot product is commutative - the order of the vectors does not matter. a⋅b=b⋅a.

Algebraic Properties of the Dot Product

(1) (Commutative Property) For any two vectors A and B, A^.B = B^.A. (2) (Scalar Multiplication Property) For any two vectors A and B and any real number c, (cA)^.B = A^.(cB) = c(A^.B) (3) (Distributive Property) For any 3 vectors A, B and C, A^.(B+C) = A^.B + A^.C.

Definition. A k-by-1 matrix is called a column vector and a 1-by-k matrix is called a row vector. The coefficients in rowi of the matrix A determine a row vector Ai =(ai1, ai2, …,ain), and the coefficients of column j of A determine a column vector Aj = ha1 j,a2 j,...,amji.

To convert a vector into matrix, just need to use matrix function. We can also define the number of rows and columns, if required but if the number of values in the vector are not a multiple of the number of rows or columns then R will throw an error as it is not possible to create a matrix for that vector.

Vectors as matrices

Because a vector is just a list of numbers, we can represent it as a matrix. This is why we can write vectors with matrix notation. We can always write a given vector as a row vector ( 1 × n ‍ ) or as a column vector ( n × 1 ‍ ).