Scalars and Vectors Worksheets
Scalars and vectors are things that we use to describe objects and the motion of those objects. We would normally use descriptive words to describe this motion like going fast and measure the velocity of that movement. We have already learned that vectors are quantities that we use to fully describe the magnitude and the direction of motion. Scalars also describe a magnitude, but not a direction. Scalars simply provide a numerical value that provides a range of the scale. These worksheets focus on understanding and using these components of a vector and scalars. The lessons show you how to add the dimension of a scale to vectors.
Aligned Standard: HSN-VM.B.5
- Find Vector Angles Step-by-step Lesson- Find the angle between two vectors.
- Guided Lesson - Find a perpendicular vectors and an angle between two vectors.
- Guided Lesson Explanation - I walk you through the complexity of the component method and using the cross product.
- Practice Worksheet - We work both skills through until we have a good handle on it.
- Matching Worksheet - The radical quotients kind of give away which problems are related to which answers.
- Answer Keys - These are for all the unlocked materials above.
Find angles, vectors, and magnitudes of vectors.
- Homework 1 - When we find the dot product of given vectors, it tells us that: A * B = Ax Bx + Ay By+ Az Bz
- Homework 2 - A perpendicular vector can be found by finding the cross product of both vectors.
- Homework 3 - Taking inverse on both sides will give the value of Θ .
Sometimes students get in the habit of pitching guesses and then working off of that. I prefer to calculate first.
- Practice 1 - Putting it all together gives the coordinates of the vector.
- Practice 2 - You can also use this to prove they have a relationship such as parallel or perpendicular.
- Practice 3 - Find a vector perpendicular to both u = (6, -5, 0) and v = (2, 6, 5).
Math Skill Quizzes
Finding perpendicular vectors seems to baffle a few students. I find that they can't define what "perpendicular" means in most cases.
- Quiz 1 - Find the angle Θ between the vectors A = (7, 2, 2) and B = (4, -9, 4)?
- Quiz 2 - See if you can crack what they are looking for here.
- Quiz 3 - The key to look for here is the angle between them.
How to Multiply a Vector by a Scalar
Vectors and algebra are quite alike, given that the coefficient changes the value of the unknown variable in algebra. While scale quantity increases the value (magnitude in vectors). Vectors are quantities that not only have a value or magnitude, but they have a direction they are going. Examples of vector quantities are measures like velocity and acceleration. Both of these would also be accompanied by a direction like North or West. Scalar quantities are those that just have a magnitude or numeric value that does not require a direction like temperature or mass.
There will be times when we need to perform various operations between quantities that are scalar or vectors. Multiplying a vector with a scalar increases the magnitude however, the direction of the vector remains the same. Let us take a few examples of multiplying a vector with a scalar quantity. Suppose you have a vector 2i + 3j - 5k, and you have to multiply the vector with the scalar 5.
5 (2i + 3j - 5k) | 10i + 15j - 25k) | It's quite simple, you just have to multiply the constant numbers or scalar quantity with the vector quantity. Geometrically, you can say that the length of the vector increases in all the dimensions.
Finding Perpendicular Vectors
There are many times that we will studying the motion of objects through the use of vectors, and we often wish to learn how to stabilize or counter that object for many different purposes. If have a given vector that you would like to construct a perpendicular vector it can be found by finding the cross product of both vectors. Vectors may be presented in two- or three-dimensional form and this need to be taken into account when you process this operation. For two dimensional vectors you would find the change in the x and y portions of the vector using:
x = uy x vz - uz x vy
y = uz x vx - ux x vz
and if you have a third dimension, you would include the z coordinate with:
z = ux x vy - uy x vx