# Write a vector expression for in unit-vector notation

Finally, for the k component we have: This gives the final result using the graphical methods is obtained by adding the three components as: Another approach to perform cross products using index notation can be developed using the cross products among the orthonormal basis vectors. In this tutorial we will examine some of the elementary ideas concerning vectors. The reason for this introduction to vectors is that many concepts in science, for example, displacement, velocity, force, acceleration, have a size or magnitude, but also they have associated with them the idea of a direction.

And it is obviously more convenient to represent both quantities by just one symbol. That is the vector. Graphically, a vector is represented by an arrow, defining the direction, and the length of the arrow defines the vector's magnitude.

This is shown in Panel 1. If we denote one end of the arrow by the origin O and the tip of the arrow by Q.

Then the vector may be represented algebraically by OQ. Panel 1 This is often simplified to just. The line and arrow above the Q are there to indicate that the symbol represents a vector. Another notation is boldface type as: Note, that since a direction is implied.

The magnitude of a vector is denoted by absolute value signs around the vector symbol: The operation of addition, subtraction and multiplication of ordinary algebra can be extended to vectors with some new definitions and a few new rules.

There are two fundamental definitions. Panel 2 2 A vector having the same magnitude as A but in the opposite direction to A is denoted by -Aas shown in Panel 3. Panel 3 We can now define vector addition.

The sum of two vectors, A and B, is a vector C, which is obtained by placing the initial point of B on the final point of A, and then drawing a line from the initial point of A to the final point of Bas illustrated in Panel 4. This is sometines referred to as the "Tip-to-Tail" method.

Use the "BACK" buttion to return to this point. Vector subtraction is defined in the following way. Thus vector subtraction can be represented as a vector addition. The graphical representation is shown in Panel 5.

Inspection of the graphical representation shows that we place the initial point of the vector -B on the final point the vector Aand then draw a line from the initial point of A to the final point of -B to give the difference C.

Panel 5 Any quantity which has a magnitude but no direction associated with it is called a "scalar". For example, speed, mass and temperature. Many of the laws of ordinary algebra hold also for vector algebra. Commutative Law for Addition: If we add A and B we get a vector E. And similarly if B is added to Cwe get F.

Stop now and make sure that you follow the above proof. Panel 6 Commutative Law for Multiplication: Vectors can be related to the basic coordinate systems which we use by the introduction of what we call "unit vectors. The quantity is read as "a hat" or "a unit". Let us consider the two-dimensional or x, y Cartesian Coordinate System, as shown in Panel 7.

We can define a unit vector in the x-direction by or it is sometimes denoted by. Similarly in the y-direction we use or sometimes. Any two-dimensional vector can now be represented by employing multiples of the unit vectors, andas illustrated in Panel 8. Where Ax and Ay are vectors in the x and y directions. The actual operation implied by this is shown in Panel 9.

Remember and or have a magnitude of 1 so they do not alter the length of the vector, they only give it its direction.

Panel 9 The breaking up of a vector into it's component parts is known as resolving a vector. Depending on the orientation of the coordinate system with respect to the vector in question, it is possible to have more than one set of components.The expression is evaluated as a function (or procedure) which takes a positive integer as an argument (that is, Vector index) and returns the value of the corresponding entry of the Vector.

table The table is referenced by index of the form [j] where j is a positive integer (that is, Vector index). It's just one way textbooks write it, and is exactly equivalent to the right-most expression, which is probably the most obvious way to write the gravitational force in vector notation.

share | . In rectangular notation, the first quantity is the “real” component (horizontal dimension of vector) and the second quantity is the “imaginary” component (vertical dimension of vector).

The imaginary component is preceded by a lower-case “j,” sometimes called the j operator. In the last chapter we saw how neural networks can learn their weights and biases using the gradient descent algorithm.

There was, however, a gap in our explanation: we didn't discuss how to compute the gradient of the cost function. That's quite a gap! In this chapter I'll explain a fast algorithm for computing such gradients, an algorithm known as backpropagation.

## Write an expression for a unit vector at 45∘ clockwise from the x-axis. Express

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A three-dimensional vector v can be specified in the following form, using unit vector notation: = ^ + ȷ ^ + ^ Where v x, v y, and v z are the scalar components of v. Scalar components may be positive or negative; the absolute value of a scalar component is a magnitude.

Unit vector notation (video) | Khan Academy