# Inner Product Of Two Vectors And Angle Between Two Vector Pdf

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In mathematics , the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers usually coordinate vectors , and returns a single number. In Euclidean geometry , the dot product of the Cartesian coordinates of two vectors is widely used.

In this chapter, it will be necessary to find the closest point on a subspace to a given point, like so:. The closest point has the property that the difference between the two points is orthogonal , or perpendicular , to the subspace. For this reason, we need to develop notions of orthogonality, length, and distance. The basic construction in this section is the dot product , which measures angles between vectors and computes the length of a vector.

## 11.3: The Dot Product

If we apply a force to an object so that the object moves, we say that work is done by the force. Previously, we looked at a constant force and we assumed the force was applied in the direction of motion of the object. Under those conditions, work can be expressed as the product of the force acting on an object and the distance the object moves. In this chapter, however, we have seen that both force and the motion of an object can be represented by vectors. In this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions.

The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes. It even provides a simple test to determine whether two vectors meet at a right angle.

We have already learned how to add and subtract vectors. In this chapter, we investigate two types of vector multiplication. The first type of vector multiplication is called the dot product, based on the notation we use for it, and it is defined as follows:. When two vectors are combined under addition or subtraction, the result is a vector. When two vectors are combined using the dot product, the result is a scalar. For this reason, the dot product is often called the scalar product.

It may also be called the inner product. The calculation is the same if the vectors are written using standard unit vectors. We still have three components for each vector to substitute into the formula for the dot product:. Like vector addition and subtraction, the dot product has several algebraic properties. We prove three of these properties and leave the rest as exercises. The associative property looks like the associative property for real-number multiplication, but pay close attention to the difference between scalar and vector objects:.

The fourth property shows the relationship between the magnitude of a vector and its dot product with itself:. Note that by property iv. Simplifying this expression is a straightforward application of the dot product:. The dot product provides a way to find the measure of this angle. This property is a result of the fact that we can express the dot product in terms of the cosine of the angle formed by two vectors.

The dot product of two vectors is the product of the magnitude of each vector and the cosine of the angle between them:. Applying the law of cosines here gives. Using this equation, we can find the cosine of the angle between two nonzero vectors. Start by finding the value of the cosine of the angle between the vectors:.

Round to the nearest hundredth. We can formalize this result into a theorem regarding orthogonal perpendicular vectors. The terms orthogonal, perpendicular, and normal each indicate that mathematical objects are intersecting at right angles. The use of each term is determined mainly by its context. We say that vectors are orthogonal and lines are perpendicular. The term normal is used most often when measuring the angle made with a plane or other surface. The angle a vector makes with each of the coordinate axes, called a direction angle, is very important in practical computations, especially in a field such as engineering.

For example, in astronautical engineering, the angle at which a rocket is launched must be determined very precisely. A very small error in the angle can lead to the rocket going hundreds of miles off course. Direction angles are often calculated by using the dot product and the cosines of the angles, called the direction cosines.

Therefore, we define both these angles and their cosines. The cosines for these angles are called the direction cosines. So far, we have focused mainly on vectors related to force, movement, and position in three-dimensional physical space. However, vectors are often used in more abstract ways. For example, suppose a fruit vendor sells apples, bananas, and oranges. On a given day, he sells 30 apples, 12 bananas, and 18 oranges. In this example, although we could still graph these vectors, we do not interpret them as literal representations of position in the physical world.

We are simply using vectors to keep track of particular pieces of information about apples, bananas, and oranges. This idea might seem a little strange, but if we simply regard vectors as a way to order and store data, we find they can be quite a powerful tool. We then add all these values together. So, in this example, the dot product tells us how much money the fruit vendor had in sales on that particular day. When we use vectors in this more general way, there is no reason to limit the number of components to three.

What if the fruit vendor decides to start selling grapefruit? In that case, he would want to use four-dimensional quantity and price vectors to represent the number of apples, bananas, oranges, and grapefruit sold, and their unit prices.

As you might expect, to calculate the dot product of four-dimensional vectors, we simply add the products of the components as before, but the sum has four terms instead of three. AAA Party Supply Store sells invitations, party favors, decorations, and food service items such as paper plates and napkins. During the month of May, AAA Party Supply Store sells invitations, party favors, decorations, and food service items. Use vectors and dot products to calculate how much money AAA made in sales during the month of May.

How much did the store make in profit? We have. To calculate the profit, we must first calculate how much AAA paid for the items sold. Their profit, then, is given by. All their other costs and prices remain the same. If AAA sells invitations, party favors, decorations, and food service items in the month of June, use vectors and dot products to calculate their total sales and profit for June.

As we have seen, addition combines two vectors to create a resultant vector. But what if we are given a vector and we need to find its component parts?

We use vector projections to perform the opposite process; they can break down a vector into its components. The magnitude of a vector projection is a scalar projection.

We return to this example and learn how to solve it after we see how to calculate projections. To find the two-dimensional projection, simply adapt the formula to the two-dimensional case:. Sometimes it is useful to decompose vectors—that is, to break a vector apart into a sum. This process is called the resolution of a vector into components. Projections allow us to identify two orthogonal vectors having a desired sum.

Then, we have. Its engine generates a speed of 20 knots along that path see the following figure. In addition, the ocean current moves the ship northeast at a speed of 2 knots. Round the answer to two decimal places.

We get. The ship is moving at Repeat the previous example, but assume the ocean current is moving southeast instead of northeast, as shown in the following figure. Now that we understand dot products, we can see how to apply them to real-life situations. The most common application of the dot product of two vectors is in the calculation of work. From physics, we know that work is done when an object is moved by a force. We saw several examples of this type in earlier chapters. Now imagine the direction of the force is different from the direction of motion, as with the example of a child pulling a wagon.

To find the work done, we need to multiply the component of the force that acts in the direction of the motion by the magnitude of the displacement. The dot product allows us to do just that. The customary unit of measure for work, then, is the foot-pound. One foot-pound is the amount of work required to move an object weighing 1 lb a distance of 1 ft straight up. Find the work done by the conveyor belt.

The distance is measured in meters and the force is measured in newtons. What is the work done by this force? Learning Objectives Calculate the dot product of two given vectors. Determine whether two given vectors are perpendicular. Find the direction cosines of a given vector. Explain what is meant by the vector projection of one vector onto another vector, and describe how to compute it. Calculate the work done by a given force. Find each of the following products.

Hint Use four-dimensional vectors for cost, price, and quantity sold. Projections As we have seen, addition combines two vectors to create a resultant vector.

## 11.3: The Dot Product

If we apply a force to an object so that the object moves, we say that work is done by the force. Previously, we looked at a constant force and we assumed the force was applied in the direction of motion of the object. Under those conditions, work can be expressed as the product of the force acting on an object and the distance the object moves. In this chapter, however, we have seen that both force and the motion of an object can be represented by vectors. In this section, we develop an operation called the dot product, which allows us to calculate work in the case when the force vector and the motion vector have different directions. The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes.

Join Stack Overflow to learn, share knowledge, and build your career. Connect and share knowledge within a single location that is structured and easy to search. What the most efficient way in the programming language R to calculate the angle between two vectors? Note: if you have x and y in matrix form, use as. It follows that the R code to calculate the angle between the two vectors is.

A vector can be multiplied by another vector but may not be divided by another vector. There are two kinds of products of vectors used broadly in physics and engineering. One kind of multiplication is a scalar multiplication of two vectors. Taking a scalar product of two vectors results in a number a scalar , as its name indicates. Scalar products are used to define work and energy relations. For example, the work that a force a vector performs on an object while causing its displacement a vector is defined as a scalar product of the force vector with the displacement vector.

## Dot product

The next topic for discussion is that of the dot product. Sometimes the dot product is called the scalar product. The dot product is also an example of an inner product and so on occasion you may hear it called an inner product. The theorem works for general vectors so we may as well do the proof for general vectors.

Given the geometric definition of the dot product along with the dot product formula in terms of components, we are ready to calculate the dot product of any pair of two- or three-dimensional vectors. Do the vectors form an acute angle, right angle, or obtuse angle? Home Threads Index About.

In mathematics , an inner product space or a Hausdorff pre-Hilbert space [1] [2] is a vector space with a binary operation called an inner product. They also provide the means of defining orthogonality between vectors zero inner product. Inner product spaces generalize Euclidean spaces in which the inner product is the dot product , [4] also known as the scalar product to vector spaces of any possibly infinite dimension , and are studied in functional analysis.

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