Properties Of Scalar / Dot Product
1. The dot product of $\overrightarrow$ and $\overrightarrow$ is maximum when $\theta $ = 0°
i.e. $\overrightarrow$.$\overrightarrow$ = a b cos0°= ab
2. A dot product of a and b is zero when if $\theta $ = 90°
i.e. $\overrightarrow$.$\overrightarrow$ = a b cos90°= 0
3. The scalar / dot product of two vectors $\overrightarrow$ and $\overrightarrow$ follows commutative law$\overrightarrow$.$\overrightarrow$ = $\overrightarrow$.$\overrightarrow$
4. The dot product of two vectors $\overrightarrow$ and $\overrightarrow$ follows distributive law. i.e. $\overrightarrow$. =$\overrightarrow$.$\overrightarrow$ +$\overrightarrow$.$\overrightarrow$
5. The square of a vector is a scalar quantity.
i.e. 2 =$\overrightarrow$.$\overrightarrow$ = a a cos0°= a2
$\therefore $ 2 = a2 is a scalar quantity.
Physics All Around Us
So, now you have some examples of scalar and vector quantities and you understand some of the differences between them. For more physics review, check out these contact force examples that you’re likely to see in the physical world. You can also delve more into the laws of physics with everyday examples of inertia.
How Does A Vector Quantity Differ From A Scalar Quantity
A scalar quantity describes strictly only the magnitude, or amount, of something. It is represented by a numerical value only and gives no other information.
A vector quantity, on the other hand, describes both the magnitude and direction of something.
When trying to differentiate between scalar and vector quantities, one must keep their definitions in mind. Is the amount given just a numerical value, or does it include a direction as well?
Some examples of scalar quantities are energy, time, volume, temperature, and speed. All of these quantities simply have a magnitude, and if not associated with a specific direction, are scalar quantities
Some vector quantities include displacement, force, and velocity . All these quantities are associated with both a magnitude and a certain direction.
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Resolution Of Vector In Physics
Just as it is possible to combine two or more vectors, it is possible to divide a vector into two or more parts. These split parts are called components of a given vector.
If a vector is divided into two or more vectors in such a way that the original vector is the resultant vector of the divided parts. Then those divided parts are called the components of the vector.
Dividing a vector into two components in the process of vector division will solve almost all kinds of problems. Thus, it goes without saying that vector algebra has no practical application of the process of division into many components.
So, below we will discuss how to divide a vector into two components.
University Physics Volume 1
- Describe the difference between vector and scalar quantities.
- Identify the magnitude and direction of a vector.
- Explain the effect of multiplying a vector quantity by a scalar.
- Describe how one-dimensional vector quantities are added or subtracted.
- Explain the geometric construction for the addition or subtraction of vectors in a plane.
- Distinguish between a vector equation and a scalar equation.
Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, a class period lasts 50 min or the gas tank in my car holds 65 L or the distance between two posts is 100 m. A physical quantity that can be specified completely in this manner is called a scalar quantity. Scalar is a synonym of number. Time, mass, distance, length, volume, temperature, and energy are examples of scalar quantities.
Figure 2.2 We draw a vector from the initial point or origin to the end or terminal point , marked by an arrowhead. Magnitude is the length of a vector and is always a positive scalar quantity.
Figure 2.3 The displacement vector from point A to point B is indicated by an arrow with origin at point A and end at point B. The displacement is the same for any of the actual paths that may be taken between points A and B.
Figure 2.4 A displacement \overset of magnitude 6 km is drawn to scale as a vector of length 12 cm when the length of 2 cm represents 1 unit of displacement .
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List Of Physical Quantities
This is a list of physical quantities.
The first table lists the base quantities used in the International System of Units to define the physical dimension of physical quantities for dimensional analysis. The second table lists the derived physical quantities. Derived quantities can be mentioned in terms of the base quantities.
Note that neither the names nor the symbols used for the physical quantities are international standards. Some quantities are known as several different names such as the magnetic B-field which known as the magnetic flux density, the magnetic induction or simply as the magnetic field depending on the context. Similarly, surface tension can be denoted by either , or T. The table usually lists only one name and symbol.
The final column lists some special properties that some of the quantities have, such as their scaling behavior , their transformation properties ” rel=”nofollow”>scalar, vector or tensor), and whether the quantity is conserved.
Example Question #: Understanding Scalar And Vector Quantities
Displacement is a vector quantity; it will have both magnitude and direction.
First we need to find his total distance travelled along the y-axis. Let’s say that all of her movement north is positive and south is negative.
. She moved a net of 30 meters to the north.;
Now let’s do the same for the x-axis, using positive for east and negative for west.
. She moved a net of 29 meters to the east.
Now, to find the resultant displacement, we use the Pythagorean Theorem. Her net movement north will be perpendicular to her net movement east, forming a right triangle. Her location relative to her starting point will be the hypotenuse of the triangle.
Now take the square root of both sides.
Since we are solving for a vector, we also need to find the direction of this distance. We do this by solving for the angle of displacement.
To find the angle, we use the arctan of our directional displacements in the x- and y-axes. The tangent of the angle will be equal to the x-displacement over the y-displacement.
Combining the magnitude and direction of our distance gives us the displacement: .
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Like And Unlike Vectors
Two vectors are said to be like vectors if their direction will be the same, whereas, in unlike vectors, the direction will be opposite to each other. The given diagram shows the difference between like and unlike vectors.
Here we can see in the above image that both the vectors A and B are in the same direction through their magnitude may or not be equal but they are called like vectors.
Multiplication Of A Vector By A Scalar:
When $\overrightarrow$ is multiplied by a scalar m the result, m$\overrightarrow$ is a vector quantity. In this case the product obtained is a new physical quantity which has the same direction as that of $\overrightarrow$.
For example, m$\overrightarrow$=$\overrightarrow$ where m is mass, $\overrightarrow$ is the acceleration and $\overrightarrow$is force.
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What Is Vector Quantity In Physics Can Be Fun For Everyone
You dont need to come across every question correct to find the most score for the test. There are 3 varieties of collisions. We are going to observe how fast you can locate it. In the later instance, its simplest to divide the vectors into components at the Draw a Picture stage so you dont have to be concerned about it as you get in the problem itself. TO better understand, let us take a look at the example given below.
Rather than being commutative, its anticommutative. This is the point where the cos comes in. Because human beings arent really very good at remembering things. For this instance, the answer of 45 degrees have to be correct.
What What Is Vector Quantity In Physics Is And What It Is Not
But theyre not thinking about how powerful momentum is and whats required to develop or maintain it. Impulse is quite a subtle and little known facet of our everyday lives. Both the direction and size of the movement are necessary to acquire the individual needing directions to the appropriate location. Without consistent reminders about your targets and priorities, it isnt hard to get distracted from your targets and priorities.
This multiplication is known as cross solution, and in different texts, you may locate terms outer merchandise and vector product. For the time being, the emphasis is upon the simple fact a force is a vector quantity which has a direction. It is a quantity that is measured using the standard metric unit known as the Newton. Visit SI units list to understand the units of some of the main units.
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Defining Scalar And Vector Quantities
Understanding the difference between scalar and vector quantities is an important first step in physics. The main difference in their definitions is:
- Scalar is the measurement of a unit strictly in magnitude.
- Vector is a measurement that refers to both the magnitude of the unit and the direction of the movement the unit has taken.
In other words, scalar quantity has magnitude, such as size or length, but no particular direction. When it does have a particular direction, it’s a vector quantity.
What Is A Vector Quantity
Lets begin with an example for better comprehension. Displacement is considered a vector quantity. Why?
Displacement refers to the shortest measurement between the original position of an object to its current position. The route of the object from the initial to the current position is usually not linear.
A car, for instance, must have traveled up, down, left, right to reach the current position. Through distance you can know the measured value, but how will you find the direction in which the car is parked right now?
It is displacement, i.e., a Vector quantity that considers both magnitude and direction to give you the shortest distance between original and current points, in addition to the direction as north, south, west, east, or any of the combinations.
Editors Choice: Difference Between Scalar and Vector Quantity
What Are Vectors In Physics
A physical quantity is a quantity whose physical properties you can measure. Such as mass, force, velocity, displacement, temperature, etc.
Suppose you are told to measure your happiness. In this case, you can never measure your happiness. That is, you cannot describe and analyze with measure how much happiness you have. So, happiness here is not a physical quantity.
Suppose you have a fever. And the doctor ordered you to measure your body temperature. Then you measured your body temperature with a thermometer and told the doctor. When you tell your doctor about your body temperature, you need to use the word degree centigrade or degree Fahrenheit.
So, the temperature here is a measurable quantity. So we will use temperature as a physical quantity.
In general, we will divide the physical quantity into three types
In this tutorial, we will only discuss vector quantity. But before that, lets talk about scalar.
What Are Scalars And Vectors
Scalars are quantities that have magnitude. These can be the temperature in your room, the amount of time you spend reading this reviewer, or even the mass of your laptop.
Scalars are often represented by a letter and are all numerical quantities that only answer the question, How much?
On the other hand, vectors are quantities that have magnitude and direction. In studying physics, these are important because we need these to understand other physics topics such as force.
When we study force, we do not only look at how much force is being applied but also in what direction. These respond to the questions, How much? and Which way?.
Vectors are also represented by an arrow where the length of the arrow corresponds to the vectors magnitude, the tail represents the origin of the vector and the arrowhead indicates the direction of the vector.;;;;;;;;
As we study physics, we must predict the motion of the objects. To do so, we should be knowledgeable of operations involving vectors. ;
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What Are Scalar And Vector Quantity
When we think about Scalar and vector the first thing comes into our mind is Science because the term scalar and vector are specified and explained in Science only. After science, the second thing that comes to our mind is about quantity because we name scalar and vector as quantities like scalar quantity and vector quantity. We think that there is nothing to say about scalar and vector as they are only quantities and as it comes under science Part.
Physical Significance Of Vector Product:
If two vector $\overrightarrow$ and $\overrightarrow$ are represented by two adjacent sides of a parallelogram both in magnitude and direction then the magnitude of $\overrightarrow$×$\overrightarrow$ gives the area of that parallelogram.
i.e. $\left| \overrightarrow\times \overrightarrow \right|$ = ab sin$\theta $ = area of parallelogram OACB.
Some important numerical problems of Vector:
Q.1 A disoriented physics professor drives 3.25 km north, then 4.75 km west and then 1.50 km south. Find the magnitude and direction of the resultant displacement.
Ans: 5.06km, 20.22o north of west .
Q.2 A spelunker is surveying a cave. She follows a passage of 180m straight west, then 210m in a direction 45o east of south, and then 280m at 30o east of north. After a fourth unmeasured displacement she finds herself back where she started. Use the method of components to determine the magnitude and direction of the fourth displacement.
Ans: 144m, 41o S of W.
Q. 3 A cave explorer is surveying a cave. He follows a passage 100m straight east, then 50m in a direction 30o west of north, then 150m at 45o west of south. After a fourth unmeasured displacement he finds himself back where he started. Using a scale drawing to determine the forth displacement .
Ans: 70.02m in the direction 26.34o east of north.
Ans: 1119N and direction 13.4o.
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Quiz: Scalar And Vector Quantity In Physics
Scalar and Vector are just two of the many quantities used in physics. Scalar is a quantity that is totally described by magnitude or size, whereas, a vector quantity is specified by both magnitudes as well as direction. So, here in this quiz, we are going to ask you nineteen questions about the same, read them carefully and answer correctly.
- Which of the following is a physical quantity that has a magnitude but no direction?
Vectors if multiplied or scalars if divided
Scalar, vector, vector
-12.5 m/s south
A baseball thrown to home plate
Vectors Scalars And Coordinate Systems
- Define and distinguish between scalar and vector quantities.
- Assign a coordinate system for a scenario involving one-dimensional motion.
What is the difference between distance and displacement? Whereas displacement is defined by both direction and magnitude, distance is defined only by magnitude. Displacement is an example of a vector quantity. Distance is an example of a scalar quantity. A vector is any quantity with both magnitude and direction. Other examples of vectors include a velocity of 90 km/h east and a force of 500 newtons straight down.
The direction of a vector in one-dimensional motion is given simply by a plus or minus sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vectors magnitude and points in the same direction as the vector.
Some physical quantities, like distance, either have no direction or none is specified. A scalar is any quantity that has a magnitude, but no direction. For example, a 20ºC temperature, the 250 kilocalories of energy in a candy bar, a 90 km/h speed limit, a persons 1.8 m height, and a distance of 2.0 m are all scalarsquantities with no specified direction. Note, however, that a scalar can be negative, such as a 20ºC temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.
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Properties Of Vector / Cross Product:
;;; i.e. $\left| \overrightarrow\times \overrightarrow \right|$ = a b sin90°= ab
;;;; i.e. $\overrightarrow\times \overrightarrow\ne \overrightarrow\times \overrightarrow$ but $\vec\times \vec=-$
;;;; 4. The vector product always obeys distributive law.
;;;; i.e. $\overrightarrow\times (\overrightarrow+\overrightarrow\text\overrightarrow\times \overrightarrow+\overrightarrow\times \overrightarrow$
;;;; 5. The magnitude of cross product of a vector with itself is zero i.e.
$\left| \overrightarrow\times \overrightarrow \right|$= a a sin0° = 0