On the slide we list some of the physical quantities discussed in the Beginner's Guide to Aeronautics and group them into either vector or scalar quantities. Of particular interest, the forces which operate on a flying aircraft, the weight , thrust , and aerodynmaic forces , are all vector quantities. The resulting motion of the aircraft in terms of displacement, velocity, and acceleration are also vector quantities.
These quantities can be determined by application of Newton's laws for vectors. The scalar quantities include most of the thermodynamic state variables involved with the propulsion system, such as the density , pressure , and temperature of the propellants. The energy , work , and entropy associated with the engines are also scalar quantities.
Vectors have magnitude and direction, scalars only have magnitude. The fact that magnitude occurs for both scalars and vectors can lead to some confusion. There are some quantities, like speed , which have very special definitions for scientists. By definition, speed is the scalar magnitude of a velocity vector.
A car going down the road has a speed of 50 mph. A surveyor measures the distance across a river that flows straight north by the following method. Starting directly across from a tree on the opposite bank, the surveyor walks m along the river to establish a baseline.
How wide is the river? A pedestrian walks 6. Privacy Policy. Skip to main content. Search for:. 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. Show Solution a. Algebra of Vectors in One Dimension Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. Example A Ladybug Walker A long measuring stick rests against a wall in a physics laboratory with its cm end at the floor.
The result reads that the total displacement vector points away from the cm mark initial landing site toward the end of the meter stick that touches the wall.
Check Your Understanding A cave diver enters a long underwater tunnel. Algebra of Vectors in Two Dimensions When vectors lie in a plane—that is, when they are in two dimensions—they can be multiplied by scalars, added to other vectors, or subtracted from other vectors in accordance with the general laws expressed by Figure , Figure , Figure , and Figure.
This solution is shown in Figure. Summary A vector quantity is any quantity that has magnitude and direction, such as displacement or velocity. Vector quantities are represented by mathematical objects called vectors. Geometrically, vectors are represented by arrows, with the end marked by an arrowhead.
The length of the vector is its magnitude, which is a positive scalar. On a plane, the direction of a vector is given by the angle the vector makes with a reference direction, often an angle with the horizontal. The direction angle of a vector is a scalar. Two vectors are equal if and only if they have the same magnitudes and directions. Parallel vectors have the same direction angles but may have different magnitudes.
When a vector is multiplied by a scalar, the result is another vector of a different length than the length of the original vector. Multiplication by a positive scalar does not change the original direction; only the magnitude is affected.
Multiplication by a negative scalar reverses the original direction. The resulting vector is antiparallel to the original vector. Multiplication by a scalar is distributive. Vectors can be divided by nonzero scalars but cannot be divided by vectors. Two or more vectors can be added to form another vector. The vector sum is called the resultant vector. We can add vectors to vectors or scalars to scalars, but we cannot add scalars to vectors.
Vector addition is commutative and associative. To construct a resultant vector of two vectors in a plane geometrically, we use the parallelogram rule. To construct a resultant vector of many vectors in a plane geometrically, we use the tail-to-head method.
Show Solution scalar. Show Solution answers may vary. What do vectors and scalars have in common? How do they differ? Show Solution parallel, sum of magnitudes, antiparallel, zero. Is it possible to add a scalar quantity to a vector quantity? Show Solution no, yes. Show Solution zero, yes. Can a magnitude of a vector be negative?
The remainder of this lesson will focus on several examples of vector and scalar quantities distance, displacement, speed, velocity, and acceleration. As you proceed through the lesson, give careful attention to the vector and scalar nature of each quantity. As we proceed through other units at The Physics Classroom Tutorial and become introduced to new mathematical quantities, the discussion will often begin by identifying the new quantity as being either a vector or a scalar.
To test your understanding of this distinction, consider the following quantities listed below. Categorize each quantity as being either a vector or a scalar. Click the button to see the answer. The example often given is of a cyclist going around a circular track at a steady speed but always changing direction. A geostationary satellite is in orbit above the Earth. It moves at constant speed but its velocity is constantly changing since its direction is always changing.
If its velocity is changing we say that it is accelerating. Remember, acceleration always means that some change is taking place! It is just over 10 minutes long and should give you a better understanding of Scalars and Vectors.
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