Let's look at this situation graphically. We won't go into this in any further depth, but we can consider a special case where the scalar product yields valuable information.Ĭonsider the case of a scalar product of a vector v with itself. When two arbitrary vectors are multiplied, the scalar product has a similar meaning, but the magnitude of the number is a little different. This explanation only works, however, for vectors of length 1. ) to differentiate from the vector product, which uses a times symbol ( )-hence the names dot product and cross product.Note that the operation should always be indicated with a dot ( As always, this definition can be easily extended to three dimensions-simply follow the pattern. The scalar product (or dot product) of two vectors is defined as follows in two dimensions. For simplicity, we will only address the scalar product, but at this point, you should have a sufficient mathematical foundation to understand the vector product as well. Two types of multiplication involving two vectors are defined: the so-called scalar product (or "dot product") and the so-called vector product (or "cross product"). Multiplication of two vectors is a little more complicated than scalar multiplication. Vector Multiplication: The Scalar (Dot) Product Solution: We can solve this problem algebraically quite easily. Practice Problem: Calculate the sum and difference ( t - u) of the vectors t = -2 i + 3 j and u = 6 i - 4 j. This representation provides more flexibility than the coordinate representation, but it is equivalent. Thus, we can add two vectors a and b as follows.Ī + b = (3 i – 2 j) + ( i + 3 j) = 3 i + i – 2 j + 3 j = 4 i + j Note that the unit vectors act almost identically to variables. Interested in learning more? Why not take an online Precalculus course? Graphically, we are adding two vectors in the unit directions to get our arbitrary vector. (0, 1) (multiplication rule for scalars and vectors).
Apply the rules of vectors that we have learned so far: For instance, consider the vector (2, 4). Note that any two-dimensional vector v can be represented as the sum of a length times the unit vector i and another length times the unit vector j. We can use scalar multiplication with vectors to represent vectors algebraically. So, to get a vector that is twice the length of a but in the same direction as a, simply multiply by 2. The only difference is the length is multiplied by the scalar. Solution: When we multiply a vector by a scalar, the direction of the product vector is the same as that of the factor. Practice Problem: Given a vector a = (3, 1), find a vector in the same direction as a but twice its length. (Recall that the location of a vector doesn't affect its value.) (Multiplication by a negative scalar reverses the direction of the vector, however.) The graph below shows some examples using c = 2. This is most clearly seen with unit vectors, but it applies to any vector. But what does this multiplication mean? As it turns out, multiplication by a scalar c has the effect of extending the vector's length by the factor c. Scalar multiplication is commutative, so. (Again, we can easily extend these principles to three dimensions.) Below is the definition for multiplying a scalar c by a vector a, where a = ( x, y). Let's start with the simplest case: multiplying a vector by a scalar. Multiplication involving vectors is more complicated than that for just scalars, so we must treat the subject carefully. These vectors are defined algebraically as follows.īefore we present an algebraic representation of vectors using unit vectors, we must first introduce vector multiplication-in this case, by scalars. For three dimensions, we add the unit vetor k corresponding to the direction of the z-axis. In the two-dimensional coordinate plane, the unit vectors are often called i and j, as shown in the graph below. In our standard rectangular (or Euclidean) coordinates ( x, y, and z), a unit vector is a vector of length 1 that is parallel to one of the axes. In this article, we will look at another representation of vectors, as well as the basics of vector multiplication.Īlthough the coordinate form for representing vectors is clear, we can also represent them as algebraic expressions using unit vectors. Use the scalar product to calculate the length of a vector.