Lecture 3: Addition of Vectors (Component Method and Dot Product)

Vocabulary flashcards covering methods of vector addition, forms, negation, subtraction, special cases, and the dot product as discussed in Lecture 3.

Addition of vectors (component method)

A computational method that converts all vectors to rectangular components, sums all x-components and all y-components, and then converts the resultant vector to polar form.

Rectangular form

Cartesian representation of a vector: A = Ax î + Ay ĵ (or the pair (Ax, Ay)).

Polar form

Representation of a vector by magnitude and angle: A = R(cos θ î + sin θ ĵ); x = R cos θ, y = R sin θ.

Tip-to-tail method

Graphical method of adding vectors by placing the tail of each successive vector at the tip of the previous one.

Negation of a vector (rectangular form)

Reverse the vector direction by negating its components: A → −A.

Negation of a vector (polar form)

Reverse direction by adding 180° to the angle: θ → θ + 180° (mod 360°); magnitude stays the same.

Subtraction of vectors

Subtract by adding the negation of the vector to subtract; group vectors as added and subtracted, then sum components (or use 180° shift in polar form).

180° apart (vectors)

Two vectors are opposite directions; resultant magnitude is the difference of magnitudes and the direction is that of the larger vector (polar form uses a 180° shift).

Same-direction addition

If two vectors point in the same direction, the resultant magnitude is the sum of magnitudes and the direction remains unchanged.

Adjoining quadrants

A special case where one pair of components cancels and the other adds; requires careful sign accounting.

Dot product (A · B)

The scalar (dot) product: A · B = |A||B| cos θ; result is a scalar.

Dot product in rectangular form

If A = Ax î + Ay ĵ and B = Bx î + By ĵ, then A · B = Ax Bx + Ay By.

Unit vectors

î and ĵ are unit vectors along the x and y axes; used to express vectors in rectangular form.

Magnitude of a vector

Length of the vector: |A| = sqrt(Ax^2 + Ay^2) or |A| = R in polar form.

Direction angle (θ)

Angle the vector makes with the +x axis; determine via tan θ = Ay/Ax with quadrant correction.

Conversion between polar and rectangular coordinates

Rectangular components: Ax = |A| cos θ, Ay = |A| sin θ; Magnitude: |A| = sqrt(Ax^2 + Ay^2); Angle: θ = arctan(Ay/Ax) with quadrant considerations.

Resultant vector (R)

The vector sum of the given vectors; its components are Rx and Ry, with magnitude |R| = sqrt(Rx^2 + Ry^2) and direction θ = arctan(Ry/Rx).