Vectors: Magnitude, Unit Vectors and Dot Product flashcards

Question 1

What is a vector, geometrically?

Accepted Answer

A directed line segment: a quantity drawn as a line with an arrowhead at one end.

Question 2

What is the initial point of a vector?

Accepted Answer

The point where the vector begins.

Question 3

What is the terminal point of a vector?

Accepted Answer

The point where the vector ends, marked by the arrowhead.

Question 4

What two things define a vector?

Accepted Answer

Its magnitude (the length of the line) and its direction (shown by the arrowhead).

Question 5

What does the magnitude of a vector mean?

Accepted Answer

The length of the line segment that represents the vector.

Question 6

How is a vector from point P to point Q written with an arrow notation?

Accepted Answer

As PQ with an arrow over it, showing it is a directed segment and not just a line.

Question 7

What is a position vector?

Accepted Answer

A vector whose initial point is the origin (0, 0); it is identified by its terminal point (a, b) and written ⟨a, b⟩.

Question 8

How do you turn a vector from C(x1, y1) to D(x2, y2) into its position vector?

Accepted Answer

Subtract the coordinates: ⟨x2 − x1, y2 − y1⟩.

Question 9

What is the magnitude of the position vector v = ⟨a, b⟩?

Accepted Answer

|v| = √(a² + b²), found with the Pythagorean Theorem or the distance formula.

Question 10

When does the section consider two vectors equal?

Accepted Answer

When they have the same magnitude and the same direction.

Question 11

What is the sum u + v of two vectors called, and how is it drawn?

Accepted Answer

The resultant vector; draw u, then start v at the terminal end of u, and u + v runs from the start of u to the end of v.

Question 12

How is u − v viewed in terms of addition?

Accepted Answer

As u + (−v): reverse the direction of v and add it to u.

Question 13

What is a scalar, in the context of scalar multiplication?

Accepted Answer

A constant (a number) by which a vector is multiplied.

Question 14

How do you multiply v = ⟨a, b⟩ by a scalar k?

Accepted Answer

kv = ⟨ka, kb⟩; each component is multiplied by k.

Question 15

What does multiplying a vector by a negative scalar do to its direction?

Accepted Answer

It reverses the direction (the result points opposite to the original vector).

Question 16

What is a unit vector?

Accepted Answer

A vector with a magnitude of 1.

Question 17

How do you find a unit vector in the direction of a nonzero vector v?

Accepted Answer

Divide v by its magnitude: v / |v|.

Question 18

How is a vector v with components a and b written in terms of i and j?

Accepted Answer

v = ai + bj, a linear combination of i and j.

Question 19

How is a vector with initial point P(x1, y1) and terminal point Q(x2, y2) written in terms of i and j?

Accepted Answer

v = (x2 − x1)i + (y2 − y1)j.

Question 20

What is the magnitude of v = ai + bj?

Accepted Answer

|v| = √(a² + b²).

Question 21

How do you add v = ai + bj and u = ci + dj?

Accepted Answer

v + u = (a + c)i + (b + d)j.

Question 22

How do you subtract u = ci + dj from v = ai + bj?

Accepted Answer

v − u = (a − c)i + (b − d)j.

Question 23

Given magnitude |v| and direction angle θ, what are the components of v?

Accepted Answer

x = |v| cos θ and y = |v| sin θ.

Question 24

What is the dot product of v = ⟨a, b⟩ and u = ⟨c, d⟩?

Accepted Answer

v · u = ac + bd: the product of the horizontal components plus the product of the vertical components.

Question 25

What formula gives the angle θ between two vectors v and u?

Accepted Answer

cos θ = (v / |v|) · (u / |u|), the dot product of the two unit vectors.

Vectors: Magnitude, Unit Vectors and Dot Product flashcards

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Vectors: Magnitude, Unit Vectors and Dot Product flashcards

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How this deck was checked

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