A scalar has magnitude only; a vector has both magnitude and direction. Think of scalars as single numbers (like distance, speed, mass, temperature) and vectors as arrows: length = magnitude, arrowhead = direction (examples: position vector r, displacement Δr, velocity v, acceleration a). Vectors can be written as magnitude + direction or in unit-vector/component form (Axî + Ayĵ + Azk̂). In 1D opposite directions show up as opposite signs; in 2D/3D you decompose a vector into components, add components to get a resultant (C = A + B → Cx = Ax + Bx, Cy = Ay + By). On the AP exam you’ll need to model vectors graphically and algebraically, use unit vectors, and apply sign conventions (Topic 1.1 ties directly into kinematics problems). For a quick review, see the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Ask: does the quantity need direction to be fully described? If yes → vector; if no → scalar. Scalars have magnitude only (e.g., distance, speed, mass, energy). Vectors need magnitude AND direction (e.g., position r, displacement Δr, velocity v, acceleration a, force). Visually model vectors as arrows: length ∝ magnitude, arrowhead shows direction (CED 1.1.A.2–1.1.A.3). Quick checklist: - Can you flip its sign to represent the opposite direction in 1D? (Yes → vector; use +/–.) - Do you have to add them tip-to-tail or by components? (Vectors add by components: Ax î + Ay ĵ.) - Is direction part of the definition (north, +x, 30° above horizontal)? (Yes → vector.) On the exam, be ready to show vector decomposition (unit vectors î, ĵ, k̂) or magnitude+direction and to use sign conventions in 1D (CED 1.1.A.4–1.1.A.5). Review examples and practice problems in the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and the Unit 1 overview (https://library.fiveable.me/ap-physics-c-mechanics/unit-1). For lots of practice, see (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Direction matters because some physical quantities need both how big they are (magnitude) and which way they point to fully describe their effect. Scalars (like distance and speed) only need magnitude—knowing “5 m” or “10 m/s” is enough. Vectors (like position, displacement, velocity, acceleration) require magnitude plus direction: 5 m north is different from 5 m east, and +3 m/s vs −3 m/s in 1-D tell you opposite directions (CED 1.1.A.1, 1.1.A.3, 1.1.A.5). Why it matters in practice: vector direction affects how vectors add (resultant), how motion changes (acceleration direction determines speeding up vs slowing down or turning), and how you break a vector into components (î, ĵ, ƙ) for problem solving (CED 1.1.A.4). On the exam you’ll need to use arrows, signs, or unit-vector notation and do component addition in multiple dimensions—these appear in both MCQs and FRs in Unit 1. For a quick refresher, see the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and keep practicing with problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Unit vector notation is just a clean way to write a vector by showing its components along the coordinate axes. Instead of an arrow picture, you write a vector as A = Ax î + Ay ĵ + Az k̂, where î, ĵ, k̂ are unit vectors (length = 1) pointing in the +x, +y, +z directions. The numbers Ax, Ay, Az are the components—how much of the vector points along each axis. How to use it: - If you know magnitude A and angle θ from +x in 2D: Ax = A cos θ, Ay = A sin θ, so A = (A cos θ) î + (A sin θ) ĵ. - If you know components, the magnitude is A = sqrt(Ax^2 + Ay^2 + Az^2) and the direction comes from those components. Unit-vector form makes vector addition and decomposition simple: add components (CED 1.1.A.4). For extra practice and exam-style problems on scalars vs. vectors and unit vectors, check the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and the AP Physics C practice set (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Short answer: no—you don't just add the numbers unless the vectors point in the same direction and use the same units. Vectors have magnitude AND direction, so you must add them vector-wise. How to add vectors (CED-aligned): - Graphical: place vectors head-to-tail and draw the resultant from tail of the first to head of the last. That gives direction and magnitude. - Component method (recommended for AP Physics C): break each vector into x, y (and z) components using unit vectors (î, ĵ, k̂). Add components separately: C = A + B → Cx = Ax + Bx, Cy = Ay + By (then C = Cx î + Cy ĵ). Magnitude |C| = sqrt(Cx^2+Cy^2), direction = arctan(Cy/Cx). - In 1D, use signs for direction (opposite directions = opposite signs). This is tested on the exam (unit 1 kinematics—Topic 1.1). Practice the component method on the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and try more problems at (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Scalars have magnitude only; vectors have magnitude and direction. Common scalar examples: distance, speed, mass, temperature, time, and kinetic energy. Common vector examples (from the CED): position r⃗, displacement Δr⃗, velocity v⃗, and acceleration a⃗. Vectors are drawn as arrows (length ∝ magnitude, arrow shows direction) and can be written in unit-vector/component form, e.g. r⃗ = Aî + Bĵ (+ Ck̂ in 3D). In 1D use signs for direction (positive/negative). Remember: distance vs displacement and speed vs velocity are classic AP traps—distance/speed are scalars, displacement/velocity are vectors (CED 1.1.A). Practice sketching arrow diagrams and converting to components (î,ĵ,k̂). For more review, see the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and do practice problems from the Unit 1 page (https://library.fiveable.me/ap-physics-c-mechanics/unit-1) or the practice set (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Draw a vector as an arrow: point the arrow in the vector’s direction and make the arrowhead show which way it points. The length of the arrow is proportional to the vector’s magnitude—you must pick a scale (for example, 1 cm = 2 m or 1 box = 5 m/s) and label it on the diagram. Place the tail where the vector starts (e.g., at the origin for a position vector) or use head-to-tail placement when adding vectors. To get components, drop perpendiculars to the axes and write A = Ax î + Ay ĵ where Ax = A cosθ, Ay = A sinθ. Opposite directions use opposite signs in 1-D. On the AP exam you’ll be expected to sketch arrows with correct direction, relative length, and use unit-vector or magnitude+direction notation (see Topic 1.1 CED). Practice sketching and decomposing vectors in the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and try more problems at (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Speed and velocity look similar because both measure how fast something moves, but the CED difference is simple: speed is a scalar (magnitude only); velocity is a vector (magnitude + direction). Speed tells you how fast—e.g., 5 m/s—with no direction. Velocity tells you how fast and where—e.g., 5 m/s east. Because velocity includes direction, two trips with the same speed can have different velocities (walking 5 m/s east vs. 5 m/s west). That matters on the exam: use distance/speed for scalar answers and displacement/velocity when direction matters (like sign conventions, vector components, or vector addition). For one-D, opposite directions use opposite signs; in 2-D you resolve into î and ĵ components. If you want a quick refresher, check the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and practice problems at (https://library.fiveable.me/practice/ap-physics-c-mechanics).

The hat (^) over i, j, k—written î, ĵ, k̂—means “unit vector”: a vector of magnitude 1 that points along a coordinate axis. In AP Physics C kinematics you use unit-vector notation to break any vector into components: r⃗ = Aî + Bĵ + Ck̂. Here î points in the +x direction, ĵ in +y, k̂ in +z, and the numbers A, B, C are the components (magnitudes) along those axes. Unit vectors show direction only (magnitude = 1), so multiplying a unit vector by a scalar gives a vector with that magnitude and direction. This is exactly what the CED calls “unit vector notation” (1.1.A.4.i) and is tested when you decompose/resultant vectors on the exam. For a short refresher, check the Topic 1.1 study guide on Fiveable (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV).

Use either the graphical or component method—both are on the CED. - Graphical (visual): draw vectors as arrows to scale. Put the tail of B at the tip of A (tip-to-tail). The arrow from the tail of A to the tip of B is the resultant (the vector sum). - Component (recommended for AP problems): break each vector into x and y components, add components, then rebuild the resultant with unit vectors. If A = A_x î + A_y ĵ and B = B_x î + B_y ĵ, then C = A + B = (A_x + B_x) î + (A_y + B_y) ĵ. Magnitude: |C| = sqrt((A_x + B_x)^2 + (A_y + B_y)^2). Direction (angle θ from +x): θ = arctan((A_y + B_y)/(A_x + B_x))—watch signs/quadrant. Example: A = 3î + 4ĵ, B = 1î + 2ĵ → C = 4î + 6ĵ, |C| = sqrt(52) ≈ 7.21, θ ≈ arctan(6/4) ≈ 56.3°. This component approach is how AP Physics C expects you to get exact results (use unit-vector notation on free-response). For a quick refresh, check the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and try practice questions (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Distance is a scalar: it only tells you how much ground was covered (magnitude only). Displacement is a vector: it gives the straight-line change in position from start to finish (has magnitude and direction). Example: if you walk 3 m east, then 4 m west, your total distance = 7 m, but your displacement = 1 m west (net change). On the AP CED this fits 1.1.A.1 and 1.1.A.3—distance and speed are scalars; position and displacement are vectors. In 1-D you show opposite directions with opposite signs (so displacement can be negative). Graphically, model displacement with an arrow whose length = magnitude and arrowhead = direction; algebraically you can write it in unit-vector/component form for 2D/3D. For a quick review, see the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics) to get more examples and AP-style questions.

Think of a vector as an arrow—to break it into x and y pieces, draw a right triangle with the vector as the hypotenuse. If a vector A has magnitude A and makes angle θ measured from the +x-axis, its components are Ax = A cosθ (along î) and Ay = A sinθ (along ĵ). In unit-vector form: A = (A cosθ) î + (A sinθ) ĵ. Tips: - Always sketch and label the angle (is it from +x, +y, or measured clockwise?). If θ is from +y, swap sin/cos: Ax = A sinθ, Ay = A cosθ. - Watch signs: if the component points left or down, make it negative (CED 1.1.A.5). - For addition, add components: Cx = Ax + Bx, Cy = Ay + By, then |C| = sqrt(Cx^2+Cy^2) and direction = arctan(Cy/Cx) (CED 1.1.A.4–.3). Practice a few problems (drawings + trig)—see the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and more practice questions (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Because vectors have direction, in one-dimensional motion we need a simple way to record that direction in math. Choosing an origin and a “positive” direction (right or up) lets you encode direction with a sign: quantities pointing the other way get a negative sign. That negative sign isn’t a magical physical force—it just tells you the vector points opposite your chosen positive axis. Why that’s useful: - It makes vector addition and kinematics algebraic: +3 m + (−5 m) = −2 m (net displacement left 2 m). - It keeps consistency for velocity and acceleration signs (e.g., v < 0 means moving left). - It lets you use unit vectors and components when you extend to 2D/3D (CED 1.1.A.4). Always state your sign convention on free-response work. Want a quick refresher? Check the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and try practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

The position vector r of a point gives its location from the origin and is a vector (has magnitude and direction). In unit-vector (component) form you write it as the sum of its x-, y-, and z-components times the unit vectors î, ĵ, k̂: r = x î + y ĵ + z k̂ Examples: - In 2D: r = x î + y ĵ - In 1D along x: r = x î (sign shows direction; opposite directions use opposite signs) Magnitude: |r| = sqrt(x^2 + y^2 + z^2). The unit vector in the direction of r is r̂ = r / |r|. Use this form whenever you need to add/subtract vectors or get components for kinematics (displacement, velocity, acceleration). This matches the CED statement 1.1.A.4 (unit vector notation) and helps on AP problems that ask for component decomposition or vector addition. For a quick review, see the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and try practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

When you add vectors that point in very different directions, you get a resultant that depends on their components—not just their magnitudes. Break each vector into x- and y-components (A = Ax î + Ay ĵ, B = Bx î + By ĵ), add those components, and rebuild the resultant: C = (Ax+Bx) î + (Ay+By) ĵ. Geometrically, you place vectors head-to-tail and draw the arrow from start to finish—that arrow is the resultant. If the vectors point nearly opposite, the resultant can be much smaller (even zero if they’re equal and opposite). If they’re perpendicular, use Pythagorean theorem for magnitude. Always include direction (angle) for a complete vector answer. This component method is exactly what the CED expects for vector addition (1.1.A.4–1.1.A.4.iii) and shows up on AP problems that ask for resultants or decompositions. For a quick refresher, see the Topic 1.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV) and try practice questions (https://library.fiveable.me/practice/ap-physics-c-mechanics).