If The Two Vectors Are Added Together, Which Drawing Shows The Correct Resultant Vector?
Components of a Vector
Vectors are geometric representations of magnitude and direction and tin be expressed as arrows in two or three dimensions.
Learning Objectives
Contrast two-dimensional and three-dimensional vectors
Primal Takeaways
Primal Points
- Vectors can be broken down into two components: magnitude and direction.
- Past taking the vector to be analyzed as the hypotenuse, the horizontal and vertical components can exist constitute by completing a right triangle. The bottom edge of the triangle is the horizontal component and the side opposite the angle is the vertical component.
- The angle that the vector makes with the horizontal can be used to calculate the length of the two components.
Key Terms
- coordinates: Numbers indicating a position with respect to some axis. Ex: [latex]\text{x}[/latex] and [latex]\text{y}[/latex] coordinates indicate position relative to [latex]\text{x}[/latex] and [latex]\text{y}[/latex] axes.
- axis: An imaginary line around which an object spins or is symmetrically arranged.
- magnitude: A number assigned to a vector indicating its length.
Overview
Vectors are geometric representations of magnitude and direction which are often represented past straight arrows, starting at ane bespeak on a coordinate axis and ending at a different point. All vectors have a length, called the magnitude, which represents some quality of involvement and then that the vector may be compared to another vector. Vectors, being arrows, too have a direction. This differentiates them from scalars, which are mere numbers without a direction.
A vector is defined by its magnitude and its orientation with respect to a fix of coordinates. It is ofttimes useful in analyzing vectors to break them into their component parts. For 2-dimensional vectors, these components are horizontal and vertical. For 3 dimensional vectors, the magnitude component is the same, but the direction component is expressed in terms of [latex]\text{x}[/latex], [latex]\text{y}[/latex] and [latex]\text{z}[/latex].
Decomposing a Vector
To visualize the process of decomposing a vector into its components, begin by drawing the vector from the origin of a set of coordinates. Next, depict a straight line from the origin along the x-centrality until the line is even with the tip of the original vector. This is the horizontal component of the vector. To find the vertical component, describe a line directly up from the finish of the horizontal vector until y'all reach the tip of the original vector. You should find you have a correct triangle such that the original vector is the hypotenuse.
Decomposing a vector into horizontal and vertical components is a very useful technique in understanding physics problems. Whenever you run into movement at an angle, you should remember of information technology as moving horizontally and vertically at the same fourth dimension. Simplifying vectors in this way can speed calculations and assistance to go along rail of the motility of objects.
Components of a Vector: The original vector, defined relative to a set up of axes. The horizontal component stretches from the start of the vector to its furthest x-coordinate. The vertical component stretches from the ten-centrality to the almost vertical point on the vector. Together, the two components and the vector form a right triangle.
Scalars vs. Vectors
Scalars are concrete quantities represented by a unmarried number, and vectors are represented past both a number and a direction.
Learning Objectives
Distinguish the difference between the quantities scalars and vectors represent
Key Takeaways
Key Points
- Scalars are physical quantities represented by a single number and no direction.
- Vectors are physical quantities that require both magnitude and direction.
- Examples of scalars include height, mass, area, and volume. Examples of vectors include displacement, velocity, and acceleration.
Key Terms
- Coordinate axes: A set of perpendicular lines which define coordinates relative to an origin. Instance: x and y coordinate axes ascertain horizontal and vertical position.
Concrete quantities tin usually be placed into two categories, vectors and scalars. These two categories are typified by what data they require. Vectors crave 2 pieces of information: the magnitude and direction. In dissimilarity, scalars require only the magnitude. Scalars can be thought of equally numbers, whereas vectors must be thought of more than like arrows pointing in a specific direction.
A Vector: An example of a vector. Vectors are ordinarily represented by arrows with their length representing the magnitude and their management represented by the direction the arrow points.
Vectors require both a magnitude and a management. The magnitude of a vector is a number for comparing ane vector to another. In the geometric interpretation of a vector the vector is represented by an arrow. The arrow has 2 parts that ascertain it. The two parts are its length which represents the magnitude and its management with respect to some fix of coordinate axes. The greater the magnitude, the longer the pointer. Physical concepts such as displacement, velocity, and acceleration are all examples of quantities that tin can be represented past vectors. Each of these quantities has both a magnitude (how far or how fast) and a direction. In lodge to specify a direction, there must exist something to which the management is relative. Typically this reference point is a set of coordinate axes similar the x-y aeroplane.
Scalars differ from vectors in that they do not have a direction. Scalars are used primarily to represent physical quantities for which a direction does not brand sense. Some examples of these are: mass, height, length, book, and surface area. Talking well-nigh the direction of these quantities has no meaning and then they cannot be expressed as vectors.
Adding and Subtracting Vectors Graphically
Vectors may be added or subtracted graphically by laying them terminate to cease on a gear up of axes.
Learning Objectives
Model a graphical method of vector addition and subtraction
Central Takeaways
Key Points
- To add vectors, lay the first i on a set of axes with its tail at the origin. Place the adjacent vector with its tail at the previous vector's head. When at that place are no more than vectors, draw a straight line from the origin to the head of the last vector. This line is the sum of the vectors.
- To subtract vectors, proceed as if adding the 2 vectors, but flip the vector to be subtracted across the axes then join it tail to caput as if adding.
- Adding or subtracting any number of vectors yields a resultant vector.
Primal Terms
- origin: The middle of a coordinate axis, defined equally existence the coordinate 0 in all axes.
- Coordinate axes: A ready of perpendicular lines which ascertain coordinates relative to an origin. Example: x and y coordinate axes define horizontal and vertical position.
Adding and Subtracting Vectors
One of the means in which representing concrete quantities every bit vectors makes analysis easier is the ease with which vectors may be added to one another. Since vectors are graphical visualizations, add-on and subtraction of vectors tin can be washed graphically.
The graphical method of vector addition is as well known as the caput-to-tail method. To start, draw a set of coordinate axes. Next, draw out the offset vector with its tail (base) at the origin of the coordinate axes. For vector addition it does not matter which vector you describe showtime since addition is commutative, but for subtraction ensure that the vector you describe first is the 1 you are subtracting from. The next step is to take the side by side vector and draw it such that its tail starts at the previous vector'south head (the pointer side). Continue to place each vector at the head of the preceding one until all the vectors you wish to add are joined together. Finally, draw a straight line from the origin to the head of the final vector in the chain. This new line is the vector outcome of adding those vectors together.
Graphical Addition of Vectors: The head-to-tail method of vector addition requires that you lay out the start vector forth a gear up of coordinate axes. Adjacent, place the tail of the next vector on the head of the get-go one. Describe a new vector from the origin to the head of the final vector. This new vector is the sum of the original two.
To decrease vectors the method is similar. Brand certain that the showtime vector you draw is the 1 to be subtracted from. Then, to decrease a vector, go along as if adding the opposite of that vector. In other words, flip the vector to be subtracted across the axes and and then join it tail to head every bit if adding. To flip the vector, simply put its head where its tail was and its tail where its caput was.
Adding and Subtracting Vectors Using Components
It is oft simpler to add or subtract vectors by using their components.
Learning Objectives
Demonstrate how to add together and decrease vectors by components
Primal Takeaways
Central Points
- Vectors tin can be decomposed into horizontal and vertical components.
- Once the vectors are decomposed into components, the components can be added.
- Adding the corresponding components of two vectors yields a vector which is the sum of the ii vectors.
Key Terms
- Component: A part of a vector. For example, horizontal and vertical components.
Using Components to Add and Subtract Vectors
Some other way of calculation vectors is to add the components. Previously, we saw that vectors can exist expressed in terms of their horizontal and vertical components. To add together vectors, merely limited both of them in terms of their horizontal and vertical components and then add the components together.
Vector with Horizontal and Vertical Components: The vector in this image has a magnitude of 10.3 units and a direction of 29.1 degrees above the ten-axis. It can be decomposed into a horizontal part and a vertical office as shown.
For case, a vector with a length of v at a 36.9 degree angle to the horizontal axis will accept a horizontal component of 4 units and a vertical component of 3 units. If nosotros were to add this to another vector of the same magnitude and direction, we would get a vector twice every bit long at the same angle. This can exist seen by calculation the horizontal components of the ii vectors ([latex]4+iv[/latex]) and the two vertical components ([latex]3+3[/latex]). These additions requite a new vector with a horizontal component of eight ([latex]4+4[/latex]) and a vertical component of 6 ([latex]iii+3[/latex]). To observe the resultant vector, but identify the tail of the vertical component at the head (arrow side) of the horizontal component and then depict a line from the origin to the head of the vertical component. This new line is the resultant vector. It should exist twice as long every bit the original, since both of its components are twice as large every bit they were previously.
To subtract vectors by components, simply decrease the two horizontal components from each other and do the same for the vertical components. Then draw the resultant vector as you did in the previous role.
Multiplying Vectors by a Scalar
Multiplying a vector by a scalar changes the magnitude of the vector but non the management.
Learning Objectives
Summarize the interaction betwixt vectors and scalars
Cardinal Takeaways
Key Points
- A vector is a quantity with both magnitude and direction.
- A scalar is a quantity with just magnitude.
- Multiplying a vector by a scalar is equivalent to multiplying the vector's magnitude past the scalar. The vector lengthens or shrinks but does not modify management.
Key Terms
- vector: A directed quantity, one with both magnitude and direction; the betwixt two points.
- magnitude: A number assigned to a vector indicating its length.
- scalar: A quantity that has magnitude but not direction; compare vector.
Overview
Although vectors and scalars represent different types of physical quantities, it is sometimes necessary for them to interact. While adding a scalar to a vector is impossible considering of their unlike dimensions in space, it is possible to multiply a vector by a scalar. A scalar, still, cannot be multiplied by a vector.
To multiply a vector by a scalar, just multiply the similar components, that is, the vector's magnitude past the scalar's magnitude. This will effect in a new vector with the same direction only the production of the two magnitudes.
Example
For example, if you have a vector A with a certain magnitude and direction, multiplying it by a scalar a with magnitude 0.5 will give a new vector with a magnitude of half the original. Similarly if yous take the number three which is a pure and unit of measurement-less scalar and multiply it to a vector, you get a version of the original vector which is iii times as long. As a more concrete example take the gravitational strength on an object. The strength is a vector with its magnitude depending on the scalar known every bit mass and its direction being down. If the mass of the object is doubled, the strength of gravity is doubled every bit well.
Multiplying vectors past scalars is very useful in physics. Nearly of the units used in vector quantities are intrinsically scalars multiplied by the vector. For example, the unit of meters per second used in velocity, which is a vector, is made up of two scalars, which are magnitudes: the scalar of length in meters and the scalar of time in seconds. In guild to make this conversion from magnitudes to velocity, one must multiply the unit of measurement vector in a particular direction by these scalars.
Scalar Multiplication: (i) Multiplying the vector [latex]\text{A}[/latex] by the scalar [latex]\text{a}=0.5[/latex] yields the vector [latex]\text{B}[/latex] which is one-half every bit long. (2) Multiplying the vector [latex]\text{A}[/latex] past 3 triples its length. (iii) Doubling the mass (scalar) doubles the force (vector) of gravity.
Unit Vectors and Multiplication by a Scalar
Multiplying a vector past a scalar is the aforementioned equally multiplying its magnitude by a number.
Learning Objectives
Predict the influence of multiplying a vector past a scalar
Cardinal Takeaways
Fundamental Points
- A unit vector is a vector of magnitude ( length ) 1.
- A scalar is a concrete quantity that tin be represented past a single number. Unlike vectors, scalars do not accept management.
- Multiplying a vector by a scalar is the same as multiplying the vector's magnitude by the number represented by the scalar.
Cardinal Terms
- scalar: A quantity which can be described past a single number, as opposed to a vector which requires a management and a number.
- unit vector: A vector of magnitude 1.
In add-on to calculation vectors, vectors tin can besides exist multiplied by constants known as scalars. Scalars are distinct from vectors in that they are represented by a magnitude but no management. Examples of scalars include an object's mass, height, or volume.
Scalar Multiplication: (i) Multiplying the vector A by 0.v halves its length. (2) Multiplying the vector A past three triples its length. (iii) Increasing the mass (scalar) increases the force (vector).
When multiplying a vector by a scalar, the direction of the vector is unchanged and the magnitude is multiplied by the magnitude of the scalar. This results in a new vector pointer pointing in the same direction as the former one but with a longer or shorter length. You can also accomplish scalar multiplication through the use of a vector'southward components. Once yous have the vector's components, multiply each of the components past the scalar to become the new components and thus the new vector.
A useful concept in the study of vectors and geometry is the concept of a unit vector. A unit of measurement vector is a vector with a length or magnitude of 1. The unit of measurement vectors are different for unlike coordinates. In Cartesian coordinates the directions are x and y usually denoted [latex]\hat{\text{x}}[/latex] and [latex]\hat{\text{y}}[/latex]. With the triangle higher up the letters referred to as a "hat". The unit vectors in Cartesian coordinates describe a circle known as the "unit circle" which has radius 1. This can be seen by taking all the possible vectors of length one at all the possible angles in this coordinate system and placing them on the coordinates. If y'all were to draw a line around connecting all the heads of all the vectors together, you would go a circle of radius one.
Position, Displacement, Velocity, and Acceleration as Vectors
Position, displacement, velocity, and acceleration can all be shown vectors since they are divers in terms of a magnitude and a direction.
Learning Objectives
Examine the applications of vectors in analyzing physical quantities
Key Takeaways
Key Points
- Vectors are arrows consisting of a magnitude and a direction. They are used in physics to represent physical quantities that also accept both magnitude and direction.
- Displacement is a physics term meaning the distance of an object from a reference point. Since the deportation contains two pieces of information: the altitude from the reference point and the direction away from the point, it is well represented by a vector.
- Velocity is defined every bit the rate of modify in time of the displacement. To know the velocity of an object one must know both how fast the displacement is changing and in what direction. Therefore it is also well represented by a vector.
- Acceleration, being the rate of change of velocity as well requires both a magnitude and a direction relative to some coordinates.
- When drawing vectors, you often do not have plenty space to describe them to the scale they are representing, so it is of import to denote somewhere what scale they are being drawn at.
Fundamental Terms
- velocity: The rate of alter of displacement with respect to alter in fourth dimension.
- displacement: The length and direction of a straight line between ii objects.
- dispatch: the rate at which the velocity of a body changes with time
Use of Vectors
Vectors tin be used to represent physical quantities. Nigh commonly in physics, vectors are used to correspond displacement, velocity, and acceleration. Vectors are a combination of magnitude and direction, and are drawn as arrows. The length represents the magnitude and the direction of that quantity is the direction in which the vector is pointing. Because vectors are constructed this way, it is helpful to clarify concrete quantities (with both size and direction) as vectors.
Applications
In physics, vectors are useful because they tin can visually correspond position, displacement, velocity and acceleration. When drawing vectors, you ofttimes do not have enough infinite to draw them to the scale they are representing, and so information technology is important to announce somewhere what scale they are being drawn at. For case, when drawing a vector that represents a magnitude of 100, 1 may describe a line that is 5 units long at a scale of [latex]\displaystyle \frac{1}{20}[/latex]. When the inverse of the scale is multiplied by the drawn magnitude, it should equal the actual magnitude.
Position and Displacement
Displacement is defined every bit the altitude, in any management, of an object relative to the position of another object. Physicists use the concept of a position vector as a graphical tool to visualize displacements. A position vector expresses the position of an object from the origin of a coordinate system. A position vector can also exist used to show the position of an object in relation to a reference point, secondary object or initial position (if analyzing how far the object has moved from its original location). The position vector is a straight line drawn from the arbitrary origin to the object. Once drawn, the vector has a length and a direction relative to the coordinate system used.
Velocity
Velocity is too divers in terms of a magnitude and direction. To say that something is gaining or losing velocity one must also say how much and in what management. For case, an aeroplane flying at 200 [latex]\frac{\text{km}}{\text{h}}[/latex] to the northeast can be represented by an vector pointing in the northeast direction with a magnitude of 200 [latex]\frac{\text{km}}{\text{h}}[/latex]. In cartoon the vector, the magnitude is only of import every bit a manner to compare two vectors of the same units. And so, if there were some other airplane flying 100 [latex]\frac{\text{km}}{\text{h}}[/latex] to the southwest, the vector pointer should exist one-half as long and pointing in the direction of southwest.
Dispatch
Acceleration, being the time rate of change of velocity, is equanimous of a magnitude and a direction, and is drawn with the same concept as a velocity vector. A value for acceleration would not exist helpful in physics if the magnitude and direction of this acceleration was unknown, which is why these vectors are important. In a free body diagram, for example, of an object falling, information technology would be helpful to use an acceleration vector nigh the object to denote its dispatch towards the basis. If gravity is the merely force interim on the object, this vector would be pointing down with a magnitude of 9.81 [latex]\frac{\text{m}}{\text{s}^2}[/latex] of 32.two [latex]\frac{\text{ft}}{\text{s}^two}[/latex].
Vector Diagram: Here is a human being walking up a hill. His management of travel is defined past the angle theta relative to the vertical axis and by the length of the pointer going up the loma. He is too being accelerated downward by gravity.
Source: https://courses.lumenlearning.com/boundless-physics/chapter/vectors/
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