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Table of Contents
Measurements are composed of a numerical value and an appropriate unit. Scalar quantities refer to measurements with no associated direction. Vector quantities have a specific direction associated with the measurement.
See the fact file below for more information on Scalars and Vectors, or you can download our 32-page Scalars and Vectors worksheet pack to utilize within the classroom or home environment.
Key Facts & Information
Significance of Scalars and Vectors
- Observations in physics can be studied by measuring physical quantities.
- These measurements can be generally classified as scalar quantities, measurements with no direction, and vector quantities, measurements with direction.
- By taking into consideration direction, we can better understand situations that involve movement, such as driving a car, riding a train, or walking up a hill.
Differentiating Scalars and Vectors
- A magnitude is a numerical value with a unit. Scalar measurements are simply magnitudes. A magnitude with a direction is a vector.
- All scalar measurements are one-dimensional, while vector measurements can be 1D, 2D, or even 3D.
- As an example, distance and displacement are scalar and vector measurements, respectively. A distance of 10 feet differs from a displacement of 10 feet East.
- Other examples are mass and weight. Mass is a scalar, it is a quantity that is independent of direction. Weight is a vector as the measurement is downwards—the direction of gravity.
- When looking at measurements, ask yourself: Will changing the direction make it any different? If you answered yes, then it is a vector quantity. If no, then it is a scalar quantity.
Components of a Vector
- As mentioned, a vector is composed of a magnitude and its associated direction. Changing the direction changes a vector.
- The notation |â| refers to the magnitude of vector â.
- That being said, getting the magnitude of a vector gives the scalar quantity.
- The direction of a vector is indicated by a CCW angle of rotation starting from the East.
- Thus, a 90° vector points North, and a 270° vector points South.
Mathematical Operations with Vectors in One Dimension
- The notation ĤAB refers to a vector Ĥ from point A to point B and HAB refers to a scalar quantity from point A to point B.
- When vector Ĥ is measured from point B to A, the corresponding notation is ĤBA. It is exactly the same as ĤAB, except it is measured in the opposite direction.
- Two vectors that are exactly 180° opposite to each other are called antiparallel, with ĤAB = – ĤBA.
- Two vectors that are perpendicular to each other are called orthogonal vectors.
- Two vectors with the same direction are parallel vectors.
- A vector multiplied by a scalar quantity is a vector quantity.
- A sum of vectors is called a resultant vector.
Mathematical Operations with Vectors in Two Dimensions
- The parallelogram rule is used to add two vectors in two dimensions.
- The two vectors are placed such that they start at the same point.
- Lines parallel to both vectors are drawn to create a parallelogram.
- The resultant vector is drawn from the origin to the corner of the parallelogram.
- The tail-to-head method is used for the addition of multiple vectors.
- A first vector is chosen, and the next vector is drawn with its origin (tail) at the end (head) of the first vector. This is done until all vectors are drawn.
- The resultant vector is then measured from the origin of the first vector to the end of the last vector.
Pythagorean Theorem
- The Pythagorean Theorem is useful for determining the magnitude of two vectors that are perpendicular.
- Do not forget to take the square root of the sums of a2 and b2 to get the hypotenuse! This step is often forgotten.
- This theorem can be used for problems with perpendicular vectors.
- Joseph goes on a hike and travels 3 km East and 4 km North. What is Joseph’s total distance traveled and displacement?
Trigonometry and the Direction of Vectors
- The trigonometric functions give the ratio of two sides of a right triangle, given an angle.
- A useful mnemonic is SOHCAHTOA
Sine: Opposite / Hypotenuse
Cos: Adjacent / Hypotenuse
Tan: Opposite / Adjacent - Multiplying the hypotenuse by sin(θ) or cos(θ) gives the value a or b, respectively.
- Given the ratio of 2 sides, inverse trig functions can give the angle.
- That being said, inverse trigonometric functions are used for determining the direction of vectors.
Introduction to Dot Product and Cross Product
- Dot products and cross products are two different ways to multiply vectors. They differ in that the dot product of two vectors gives a scalar quantity and a cross product gives a whole new vector. They’re denoted by their own notation.
Dot Product: î · ĵ
Cross Product: î × ĵ
- How does a dot product make a scalar quantity out of two vectors? They do this by multiplying only the x-component of the vectors. This is seen in its formula:
â · ê = |â| x |ê| x cos(θ),
where θ is the angle between the two vectors
- On the other hand, a cross-product produces a whole new vector that is perpendicular to the two original vectors. It does this by taking into account the magnitude of the vectors in all three dimensions, which requires complicated mathematics.
- The dot product is especially useful in situations where a vector is projected onto another vector. It could also be used to determine the angle between two vectors, by using the inverse cosine function on the cross product.
- The cross product, on the other hand, has multiple uses in situations that involve rotating objects, such as a mechanic turning a wrench to tighten a bolt.
Scalar and Vector Quantities in Everyday Life
- Scalar and vector quantities are present everywhere. With your newfound knowledge, hopefully, you get to better appreciate them wherever you look!
- When you throw a ball, it’s the vector sum of all forces acting on the ball which determines its velocity and acceleration as it flies through the air. And as it lands, the displacement between the origin and its final resting position is a vector quantity.
- Engineers regularly use scalar and vector quantities, from creating cars to building bridges. Through the use of free-body diagrams, they focus on an object and determine the vector sum of all forces. The net force then determines whether a car is able to move forward or whether a bridge is strong enough to carry everyone.
- Even electricity has scalar and vector quantities! Charged particles are defined through a scalar quantity, and when they move across space, they are measured as a current, a vector quantity.
- Healthcare also utilizes scalars and vectors, such as in the use of electrocardiograms (ECG). The electrocardiogram measures the action potential that courses through our heart in various directions, and this tells doctors whether a heart is healthy or sick.
- Whatever field you may be interested in, it is important to build a strong foundation in scalar and vector quantities. A good understanding of these important physics concepts will surely lead you in the right direction.
Scalars and Vectors Worksheets
This is a fantastic bundle that includes everything you need to know about Scalars and Vectors across 32 in-depth pages. These are ready-to-use worksheets that are perfect for teaching kids about Scalars and Vectors, which are quantities of measurements with a numerical value and an appropriate unit.
Complete List of Included Worksheets
Below is a list of all the worksheets included in this document.
- Scalars and Vectors Basics
- True or False
- Matching Measurements
- Everyday Physics
- Situational Identification
- Drafting Vector Sums
- Dissecting Vectors
- Vectors from Scratch
- Computing Vector Sums
- Distance and Displacement
- Free Body Diagram
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Link will appear as Scalars and Vectors Facts & Worksheets: https://kidskonnect.com - KidsKonnect, October 26, 2022
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