【People's Education Press】High School Mathematics, Compulsory Course Second Semester (Version A): Unwavering Commitment: From Physical Vectors to Geometric Representation of Plane Vectors

In physics, we often encounter quantities such as displacement, force, and velocity, which have both magnitude and direction. In mathematics, we call quantities that possess both magnitude and directionvectors (vector), while quantities with only magnitude and no direction (such as mass, time, and length) are calledscalars(in physics, known as scalars).

Geometric Representation and Fundamental Concepts of Vectors

To intuitively study vectors, we use directed line segments, namelydirected line segments (directed line segment) to represent it. A directed line segment includes three elements: initial point, direction, and length.

Magnitude of a vector: The size of vector $\vec{AB}$ is called the magnitude (or modulus) of the vector, denoted as $|\vec{AB}|$.
Special vectors: A vector with length 0 is calledthe zero vector (zero vector), denoted as $\mathbf{0}$; a vector with length equal to one unit is calleda unit vector (unit vector).
Positional relationship: Non-zero vectors with the same or opposite directions are calledparallel vectors (parallel vectors), also known ascollinear vectors (collinear vectors). It is stipulated that $\mathbf{0}$ is parallel to any vector.

The essence of vectors lies in 'freedom from positional constraints'. As long as their magnitudes are equal and their directions are the same, regardless of their starting points, they areequal vectors.

$$\boldsymbol{a} = \boldsymbol{b} \iff |\boldsymbol{a}| = |\boldsymbol{b}| \text{ and the direction is the same}$$

QUESTION 1

In physics, vectors are commonly referred to as vectors, and scalars as scalars. Which of the following quantities are vectors?

Tension on a hanging object, friction, acceleration

Pressure, frequency, mass

Temperature, altitude, density

Work, power, kinetic energy

QUESTION 2

Which of the following statements about the zero vector is correct?

The zero vector has no direction

The zero vector has a length of 0, and its direction is arbitrary

The zero vector is not parallel to any vector

All unit vectors are equal

QUESTION 3

Determine whether the following statements are true: (1) If $|\boldsymbol{a}| = |\boldsymbol{b}|$, then $\boldsymbol{a} = \boldsymbol{b}$; (2) A vector with direction south-west $60^{\circ}$ and a vector with direction north-east $60^{\circ}$ are collinear vectors.

(1) Incorrect, (2) Correct

(1) Correct, (2) Incorrect

Both are correct

Both are incorrect

QUESTION 4

Given that $\boldsymbol{a}$ and $\boldsymbol{b}$ are both unit vectors, which of the following conclusions is correct?

$\boldsymbol{a} = \boldsymbol{b}$

$|\boldsymbol{a}| = |\boldsymbol{b}| = 1$

$\boldsymbol{a} // \boldsymbol{b}$

$|\boldsymbol{a}|^2 = \boldsymbol{b}$

QUESTION 5

In rectangle $ABCD$, $AB = 2$, $BC = 1$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. Among the vectors with endpoints at $A, B, C, D, M, N$, how many are equal to $\vec{AM}$?

QUESTION 6

Example 1: In Figure 6.1-4, the scale is 1:8,000,000. If the directed line segment from A to B on the map is 2 cm long, what is the actual displacement distance?

16 km

160 km

80 km

1600 km

QUESTION 7

If two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ satisfy $\boldsymbol{a} // \boldsymbol{b}$, then:

Their endpoints must lie on the same straight line

They must have the same direction

Their directions are the same or opposite

Their lengths must be equal

QUESTION 8

Identify the lengths of the vectors in the diagram. (Small grid side length is 0.5) If vector $\vec{v}$ spans 4 grids horizontally, its magnitude is:

0.5

QUESTION 9

Determine: Is the x-axis on the Cartesian plane a vector?

Yes, because it has direction

No, because it has direction but no definite length (it is a line)