Physics: An Introduction Chapter 3 -- Reference Tools & Resources

Chapter 3: Vectors in Physics
Reference Tools & Resources

Reference Tools & Resources

I. Key Terms and Phrases

vector: a mathematical quantity having both magnitude and direction with appropriate units.

scalar: a numerical value with appropriate units.

magnitude of a vector: the full numerical value of the quantity being represented.

direction of a vector: the orientation within a coordinate system of the quantity being represented.

component of a vector: the part of a vector associated with a specific direction.

unit vector: a dimensionless of unit magnitude.

II. Important Equations

Name/Topic Equation Explanation
scalar components of a vector V Scalar components of a vector from its magnitude and direction. The angle q is measured from the +x direction.

reference angle The reference angle gives the direction of a vector with respect to the coordinate axes.

magnitude of a vector The magnitude of a vector from its x- and y-components.

unit vectors A two-dimensional vector written in terms of unit vectors.

vector addition/subtraction Vector addition and subtraction with components.

relative motion v_PA = v_PB + v_BA Velocity addition for relative motion using the subscripting mnemonic.

Name/Topic	Equation	Explanation
scalar components of a vector V		Scalar components of a vector from its magnitude and direction. The angle q is measured from the +x direction.
reference angle		The reference angle gives the direction of a vector with respect to the coordinate axes.
magnitude of a vector		The magnitude of a vector from its x- and y-components.
unit vectors		A two-dimensional vector written in terms of unit vectors.
vector addition/subtraction		Vector addition and subtraction with components.
relative motion	v_PA = v_PB + v_BA	Velocity addition for relative motion using the subscripting mnemonic.

III. Know Your Units

Quantity Dimension SI Unit
unit vector -- dimensionless

Quantity	Dimension	SI Unit
unit vector	--	dimensionless

IV. Other Useful Tips

The component method of vector addition described in this chapter is very powerful and always works. However, the method can sometimes be cumbersome and the more calculations you do provide greater opportunities for some sort of mistake to slip in. For cases when you know the magnitudes and direction of the two vectors you need to add, the law of cosines can provide a more direct route to the result than the components method. The law of cosines is illustrated below.

Consider the many calculations in the solution of example 3.4. If we only needed the magnitude of v₃, having been given the magnitude and direction of both v₁ and v₂ would have allowed us to immediately write down the triangle

The law of cosines would then yield the result with just one additional step.

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