Distinguish scalar and vector quantities, represent vectors as scaled arrows, and find the resultant of vectors by graphical and component methods, including resolving a vector into perpendicular components.

What this topic is asking

The Physical Setting/Physics Core Curriculum opens mechanics by asking you to separate scalar quantities, which have magnitude only, from vector quantities, which have both magnitude and direction, and then to combine vectors correctly. On the Regents this surfaces as quick multiple-choice items ("which of these is a vector?") and as constructed-response problems that ask for the magnitude and direction of a resultant or the components of a single vector. Getting the distinction right is the foundation for every later mechanics topic, because displacement, velocity, acceleration, force and momentum are all vectors.

Scalars and vectors

The difference matters because vectors do not add like ordinary numbers. Walking $6$ km east and then $8$ km north leaves you $10$ km from the start, not $14$ km, because the directions are different. Whenever a Regents problem mentions a direction, or asks "how far from the start" rather than "how far travelled", you are working with vectors.

Representing vectors as arrows

A vector is drawn as an arrow on a chosen scale: its length is proportional to the magnitude, and it points in the direction of the quantity. A velocity of $20$ m/s east and one of $40$ m/s east are parallel arrows, the second twice as long. Two velocities of equal speed in opposite directions are equal-length arrows pointing opposite ways, and they are different vectors even though their speeds (the scalars) are equal.

Adding vectors head to tail

For two perpendicular vectors (a very common Regents setup, such as a boat crossing a river or a hiker turning a right angle), the resultant is the hypotenuse of a right triangle:

where $\theta$ is measured from vector $A$. For two vectors at a general angle, you add them by components (below) or, on a diagram, by careful head-to-tail construction and measurement.

Resolving a vector into components

The reverse move, used constantly in projectile and force problems, is to split one vector into two perpendicular components. With the angle $\theta$ measured from the horizontal:

The horizontal component uses cosine and the vertical component uses sine when the angle is taken from the horizontal. Resolving turns an awkward diagonal vector into two independent one-dimensional problems, which is exactly how projectile motion and inclined-plane forces are handled.

To add several vectors by components, resolve each into $x$ and $y$ parts, add the $x$ parts and the $y$ parts separately to get $R_x$ and $R_y$, then recombine: $R = \sqrt{R_x^2 + R_y^2}$ with direction $\theta = \tan^{-1}(R_y/R_x)$.

Reference Tables note

The Reference Tables for Physical Setting/Physics give you the constants and the kinematics, force and energy equations, but they do not print the vector formulas $R = \sqrt{A^2 + B^2}$, $A_x = A\cos\theta$ or $A_y = A\sin\theta$. You are expected to know the Pythagorean theorem and basic right-triangle trigonometry and apply them. The tables do include the trigonometric definitions of sine, cosine and tangent for reference.

Try this

Q1. State one example each of a scalar and a vector quantity, other than those used above. [2 points]

• Cue. A scalar example: temperature (or distance, speed, mass). A vector example: weight (or any force, acceleration, momentum).

Q2. A car drives $30$ m east then $40$ m west. Calculate the magnitude of its resultant displacement. [2 points]

• Cue. These are along one line, so subtract: $40 - 30 = 10$ m west. (The distance travelled is $70$ m, but displacement is the net vector.)