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Vectors in Two and Three Dimensions

Part 2: The Dot Product

Since addition and subtraction are possible, one might wonder if it is also possible to multiply two vectors. The most obvious idea -- just multiplying corresponding components -- turns out not to be all that useful. However, the following definition has geometric significance that makes it very useful. We will explore this geometry in what follows.

Definition: The dot product of two vectors = (a,b) and w = (c,d), which we write as <v,w>, is ac + bd.

The dot product of two vectors is a scalar. For this reason the dot product is sometimes called the scalar product. The name "dot product" comes from the fact that another common notation for <v,w> is v.w.

If v and w are the same vector, we find

<v,v> = a+ b2 = |v|2.

Thus, the length of a vector is the square root of its dot product with itself.

  1. Using the applet below, experiment with the dot product of two vectors in the plane. Here, the white vector is dotted with the green vector, and the corresponding component of the green vector in the direction of the white is shown in red when you press the "projection" button. [Each white vector is fixed, but the green vector can be moved by dragging. To get a new white vector, click "Change vector".] Record any significant observations in your worksheet.

  1. Compute the dot product of v = (-3,5) with each of the vectors i = (1,0) and j = (0,1). What information is the dot product extracting from v in this case? Prove your statement for the dot product of a general vector (a,b) with either i or j.

  2. Let be the angle between two vectors v and w. Use the geometry of the triangle formed by these two vectors and their difference vector - w to express the cosine of in terms of the lengths of the vectors and their dot product. [Need a hint? Point to the Help button at the right.]

We see that the dot product tells us about the relative direction of the two vectors. If one of the vectors happens to lie on the horizontal or vertical axis of the plane, this direction is relative to our coordinate system.

  1. What does your result from the preceding step say about the dot product of two perpendicular, non-zero vectors v and w? How does this result compare to what you know about the slopes of perpendicular line segments? [Hint: What is the slope of the vector = (a,b)?]

We can use the vectors i and j and the properties of scalar multiplication and vector addition to write vectors in another way. For example, the vector (4,5) can be written as

(4,5) = 4(1,0) + 5(0,1) = 4+ 5j

  1. Write the vectors (-1,3), (10,-3) and (0,8) as sums of multiples of i and j.

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