Vector Operations and Components - Study Notes

Vector addition and subtraction

• Vector addition is associative and commutative in graphical form: the order of adding multiple vectors does not affect the final result.
  • Tail-to-head construction: to add vectors A and B, slide B so its tail starts at A's head. The resultant is the vector from A's tail to B's head.
  • For three vectors A, B, C: (A + B) + C gives the same resultant as A + (B + C).
  • If you draw A + B first, you get a red A+B; then you add C to that result to obtain the final net vector R; you can also add B + C first and then A, ending up with the same R.
• Collinear vectors: when vectors are collinear (pointing in the same or opposite direction), their sum lies on the same line; you effectively perform one-dimensional addition (sum if same direction, difference if opposite).
• Practical view: you can add any number of vectors graphically by keeping a head-to-tail chain, and the end-to-end vector is the total sum.

Subtracting vectors

• Subtraction is performed by adding a negative: A − B = A + (−B).
  • Geometric interpretation: if B points down-left, then −B points up-right; adding −B to A gives the resultant of the subtraction.
  • You can still compose the vectors head-to-tail or by placing −B after A, or equivalently place A after B with a sign convention; the final vector is unique.
• Practical note: the operation is the same whether you do A − B as (A + (−B)) or (−B + A); the order changes the intermediate drawing but not the final resultant.
• Relevance: subtraction is used for velocity differences, relative motion, and in upcoming topics like velocity and acceleration vectors.

Multiplying a vector by a scalar

• If c is a scalar, the product cA has magnitude $|cA| = |c| \, |A|.$
• Direction:
  • If c > 0, cA points in the same direction as A.
  • If c < 0, cA points in the opposite direction to A, with magnitude $|c| \, |A|.$
• Examples:
  • 2A is twice as long as A, same direction.
  • −3A has magnitude 3|A| and points opposite to A.

Trigonometry and right triangles (basis for vector components)

• Right triangle setup: angle θ, with opposite, adjacent, and hypotenuse sides.
• Sine, cosine, and tangent definitions:
  • $ext{sin}( heta) = \frac{ ext{opposite}}{ ext{hypotenuse}}$
  • $ext{cos}( heta) = \frac{ ext{adjacent}}{ ext{hypotenuse}}$
  • $ext{tan}( heta) = \frac{ ext{opposite}}{ ext{adjacent}}$
• Important note on orientation: angles and sides are defined relative to the axis chosen for θ; sometimes θ is given relative to the horizontal (x-axis) or relative to the vertical (y-axis). The identification of opposite/adjacent depends on how θ is defined.
• Practical implication: depending on how θ is measured, the components may be expressed with cos or sin accordingly (see components section).

Components of vectors in 2D and 3D

• In 2D, a vector A can be expressed by its components along x and y: $\boldsymbol{A} = (A<em>x, A</em>y)$.
• If θ is the angle of A measured from the horizontal (x-axis):
  • $A_x = |A| \, \cos\theta$
  • $A_y = |A| \, \sin\theta$
• Quadrant signs (for a standard coordinate system with x positive to the right and y positive up):
  • Quadrant I: Ax > 0, Ay > 0
  • Quadrant II: Ax < 0, Ay > 0
  • Quadrant III: Ax < 0, Ay < 0
  • Quadrant IV: Ax > 0, Ay < 0
• If θ is given relative to the vertical (y-axis), the components swap roles in the sense that:
  • $A_x = |A| \sin\theta$
  • $A_y = |A| \cos\theta$
• In cases where the angle is near the axis or in different orientations, the cosine and sine assignments can swap accordingly; the sign is determined by the quadrant, not just the trigonometric value.
• Practical tip: always identify which axis the angle is measured from, and use the quadrant to assign signs to the components.
• 3D extension (brief): for a vector in 3D,
  • $\boldsymbol{A} = (A<em>x, A</em>y, A<em>z)$ with magnitude $|A| = \sqrt{A</em>x^2 + A<em>y^2 + A</em>z^2}$ and direction cosines relative to the axes; components along each axis follow the same projection idea.

Determining components from a given vector and angle

• Example scenario (two axes): Northeast direction is exactly 45° relative to the cardinal directions (x and y). If the magnitude is |A|, then the components are:
  • $A_x = |A| \cos 45^{\circ}$
  • $A_y = |A| \sin 45^{\circ}$
• Note: for a vector whose axis orientation is not standard (e.g., axis rotated or a nonstandard angle reference), you must determine which axis the angle references and apply the corresponding cosine/sine relation.
• Important: always set the calculator to the correct angle mode (degrees or radians) matching the given angles.
• Example for a vector in a nonstandard coordinate system (rotated axis): if the x-axis is oriented down-left, you compute components with the same trigonometric relations but interpret signs based on the quadrant. In the example: dx = |d| cos α and dy = |d| sin α with signs corresponding to quadrant, which may place dy negative in QIV.
• Summary rule:
  • If the angle α is measured from the +x axis, then $d<em>x = |d| \cos\alpha, \quad d</em>y = |d| \sin\alpha.$
  • If the angle α is measured from the +y axis, then $d<em>x = |d| \sin\alpha, \quad d</em>y = |d| \cos\alpha.$
  • Always determine the quadrant to assign the correct signs to each component.

Magnitude from components and determining the direction

• Magnitude from components: $|\boldsymbol{v}| = \sqrt{v<em>x^2 + v</em>y^2}.$
• Direction (angle) from components: $\theta = \tan^{-1}\left(\frac{v<em>y}{v</em>x}\right)$ (more robustly, use $\theta = \operatorname{atan2}(v<em>y, v</em>x)$ to get the correct quadrant).

Adding and subtracting vectors using components

• Given vectors A, B, (and C, if present):
  • Compute each vector’s components: A = (Ax, Ay), B = (Bx, By) (and C = (Cx, Cy)).
  • Resultant components: $R<em>x = A</em>x + B<em>x (+ C</em>x), \quad R<em>y = A</em>y + B<em>y (+ C</em>y).$
  • Reconstruct the resultant vector: $\boldsymbol{R} = (R<em>x, R</em>y)$ with magnitude and angle derived from the above formulas.
• Example workflow for a practice problem:
  • Vector A: magnitude |A| with angle relative to the vertical (given). Convert to components: determine Ax and Ay using the appropriate cosine/sine combination for the given angle reference and quadrant.
  • Vector B: magnitude 57.3, angle is below the horizontal with 36°; compute Bx = |B| cos 36°, By = |B| sin 36° with signs determined by quadrant (here, below the horizontal implies negative y component).
  • Vector C: given as pointing downward (specify its components accordingly).
  • Sum components: Rx = Ax + Bx + Cx, Ry = Ay + By + Cy.
  • Then compute the resultant magnitude |R| = \sqrt{Rx^2 + Ry^2} and direction θR = atan2(Ry, R_x).
• Practical note: in the exam, you will often compute all x- and y-components first, then combine them to get the resultant; verify signs via the quadrant logic.

Handy reminders and exam tips

• Always check the angle mode on your calculator: degrees vs radians.
• When angles are given relative to a nonstandard axis, explicitly determine the component formulas and the sign from the quadrant.
• Use the proper vector notation: components (Ax, Ay, A_z) and the magnitude |A| with the corresponding angle direction.
• Remember the physical interpretations: components are projections of the vector onto the coordinate axes; the magnitude is the length of the vector.
• For nonzero, non-collinear vectors, use the x- and y-component method to avoid mistakes with head-to-tail drawings, especially when dealing with multiple vectors.
• For collinear vectors, compare magnitudes along the common line; if directions agree, add magnitudes; if opposite, subtract magnitudes and keep the sign/direction of the larger magnitude.
• The material indicates connections to future topics: velocity and acceleration vectors will rely heavily on these same vector operations.

Quick practice template (fill-in-the-blank steps)

• Given vectors A, B, and C with magnitudes and directional angles:
  • Step 1: Determine components for each vector: Ax, Ay; Bx, By; Cx, Cy.
  • Step 2: Compute resultant components: Rx = Ax + Bx + Cx, Ry = Ay + By + Cy.
  • Step 3: Compute magnitude: $|\boldsymbol{R}| = \sqrt{R<em>x^2 + R</em>y^2}.$
  • Step 4: Compute direction: $\theta<em>R = \operatorname{atan2}(R</em>y, R_x).$
  • Step 5: Optionally, verify with a graphical check (head-to-tail) if time permits.

Note on notational consistency

• Throughout, we represent vectors in bold (e.g., \boldsymbol{A}) and their components as Ax, Ay. Magnitudes are denoted by |A|.
• Equations are provided in LaTeX format for accuracy and to mirror standard physics/engineering notation.