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.
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