m01-2: vectors

3-1 Vectors and Their Components (1 of 15) Link to heading

Learning Objectives

3.01 Add vectors by drawing them in head-to-tail arrangements, applying the commutative and associative laws.
3.02 Subtract a vector from a second one.
3.03 Calculate the components of a vector on a given coordinate system, showing them in a drawing.
3.04 Given the components of a vector, draw the vector and determine its magnitude and orientation.
3.05 Convert angle measures between degrees and radians.

3-1 Vectors and Their Components (2 of 15) Link to heading

Physics deals with quantities that have both size and direction
A vector is a mathematical object with size and direction
A vector quantity is a quantity that can be represented by a vector
- Examples: position, velocity, acceleration
- Vectors have their own rules for manipulation
A scalar is a quantity that does not have a direction
- Examples: time, temperature, energy, mass
- Scalars are manipulated with ordinary algebra

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The simplest example is a displacement vector
If a particle changes position from A to B, we represent this by a vector arrow pointing from A to B.

Gambar (a) tiga vector A –> B, A’ –> B’ dan A" –> B" dengan arah dan besar yang sama, akan tetapi posisi A, A’, A" dan B, B’, B" tidak berhimpit.

Gambar (b) Vector A –> melalui path berbeda: langsung (merah), hurus s (kelabu biru), setengah lingkaran dan s (merah muda keputihan)

Figure 3-1

In (a) we see that all three arrows have the same magnitude and direction: they are identical displacement vectors.
In (b) we see that all three paths correspond to the same displacement vector. The vector tells us nothing about the actual path that was taken between A and B.

3-1 Vectors and Their Components (4 of 15) Link to heading

The vector sum, or resultant
- Is the result of performing vector addition
- Represents the net displacement of two or more displacement vectors $$\tag{3-1} \vec{s} = \vec{a} + \vec{b}, $$
- Can be added graphically as shown:

3-1 Vectors and Their Components (5 of 15) Link to heading

Gambar (a) Vector A –> B Actual path rounded zigzag, vector B –> C, vector A –> C Net displacement is vector sum.

Gambar (b) $\vec{a}$, $\vec{b}$, $\vec{c}$, To add $\vec{a}$ and $\vec{b}$ drawm them head to tail. This is the resulting vector, from tail of $\vec{a}$ to head of $\vec{b}$.

Figure 3-2

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Vector addition is commutative
- We can add vectors in any order $$\tag{3-2} \vec{a} + \vec{b} = \vec{b} + \vec{a} $$ (commutative law).

Gambar $\vec{a}$ ke kanan atas, disambung $\vec{b}$ ke kanan bawah; $\vec{b}$ ke kanan bawah, disambung $\vec{a}$ ke kanan atas; resultannya mendatar, Start –> Finish dengan catatan di atas garisnya $\vec{a} + \vec{b}$ dan di bawah garisnya $\vec{b} + \vec{a}$.

Keterangan: You get the same vector result for either order of adding vectors.

Figure 3-3

3-1 Vectors and Their Components (7 of 15) Link to heading

Vector addition is associative
- We can group vector addition however we like $$\tag{3-3} (\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c}) $$ (associative law).

Keterangan: You get the same vector result for any order of adding vectors.

Gambar kiri: resultan $\vec{a} + (\vec{b} + \vec{c})$ ke kanan bawah, $\vec{a}$ ke kanan atas, tegak ke bawah $\vec{b} + \vec{c}$.

Gambar tengah: $\vec{a}$ kanan atas, $\vec{b}$ kanan bawah, keduanya bersatu dengan $\vec{a} + \vec{b}$, $\vec{c}$ ke kiri bawah, resultannya $\vec{a} + \vec{b} + \vec{c}$ ke kanan bawah, tegak ke atas $\vec{b} + \vec{c}$.

Gambar kanan: ke kanan atas $\vec{a} + \vec{b}$, ke kiri bawah $\vec{c}$, dan resultannya $(\vec{a} + \vec{b}) + \vec{c}$.

3-1 Vectors and Their Components (8 of 15) Link to heading

A negative sign reverses vector direction
$$\vec{b} + (-\ve{b}) = 0.
We use this to define vector substraction $$\tag{3-4} \vec{d} = \vec{a} - \vec{b} = \vec{a} + (-\vec{b}) $$

Gambar vector $\vec{b}$ dan $-\vec{b}$, sejajar tetapi berlawanan arahnya.

Figure (3-5)

Gmabar (a) vektor $\vec{a}$ dan $\vec{b}$, $-\vec{b}$.

Gambar (b) vektor $\vec{a}$ dan $\vec{b}$, resultannya $\vec{d} = \vec{a} - \vec{b}$.

Note head-to-tail arrangement for addition (of $\vec{a}$ with $-\vec{b}$).

Figure (3-6)

3-1 Vectors and Their Components (9 of 15) Link to heading

These rules hold for all vectors, whether they represent displacement, velocity, etc.
Only vectors of the same kind can be added
- (distance) + (distance) makes sense
- (distance) + (velocity) does not

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Checkpoint 1
The magnitudes of displacements $\vec{a}$ and $\vec{b}$ are 3 m and 4 m, respectively, and $\vec{c} = \vec{a} + \vec{b}$. Considering various orientations of $\vec{a}$ and $\vec{b}$, what are (a) the maximum possible magnitude for $\vec{c}$ and (b) the minimum possible magnitude?

Answer:
(a) 3 m + 4 m = 7 m
(b) 4 m − 3 m = 1 m

3-1 Vectors and Their Components (11 of 15) Link to heading

Rather than using a graphical method, vectors can be added by components o A component is the projection of a vector on an axis
- The process of finding components is called resolving the vector
The components of a vector can be positive or negative.
They are unchanged if the vector is shifted in any direction (but not rotated).

Gambar vector $\vec{b} = (7 \ \hat{x} - 5 \ \hat{y}) \ {\rm m}$, digambarkan komponen-komponennya $b_x$ pada arah $x$ dan $b_y$ pada arah $y$, “This is the $x$ component of the vector.”, “This is the $y$ component of the vector.”, kedua sumbu dengan satuan m, grid kotak berukuran 1 m × 1 m.

Figure (3-8)

3-1 Vectors and Their Components (12 of 15) Link to heading

Components in two dimensions can be found by: $$\tag{3-5} a_x = a \cos \theta, \ \ \ \ {\rm and} \ \ \ \ a_y = a \sin \theta, $$
Where $θ$ is the angle the vector makes with the positive $x$ axis, and a is the vector length
The length and angle can also be found if the components are known $$\tag{3-6} a = \sqrt{a_x^2 + a_y^2} \ \ \ \ {\rm and} \ \ \ \ \tan \theta = \frac{a_y}{a_x} $$
Therefore, components fully define a vector

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In the three-dimensional case we need more components to specify a vector
- $(a, \theta, \phi)$ or $(a_x, a_y, a_z)$

3-1 Vectors and Their Components (14 of 15) Link to heading

Checkpoint 2
In the figure, which of the indicated methods for combining the $x$ and $y$ components of vector aԦ are proper to determine that vector?

Gambar vektor-vektor selalu membentuk segitiga

(a) $a_x < 0$, $a_y < 0$, vektro $a$ SE. (ax, ay from origin)

(b) $a_x < 0$, $a_y < 0$, vektro $a$ NW. (ax, ay from origin)

(c) $a_x < 0$, $a_y < 0$, vektro $a$ SW. (left then down)

(d) $a_x < 0$, $a_y < 0$, vektro $a$ SW. (down then left)

(e) $a_x < 0$, $a_y < 0$, vektro $a$ NE. (left then down)

(e) $a_x < 0$, $a_y < 0$, vektro $a$ SW. (all from origin)

Answer: choices (c), (d), and (f) show the components properly arranged to form the vector

3-1 Vectors and Their Components (15 of 15) Link to heading

Angles may be measured in degrees or radians
Recall that a full circle is 360 °, or 2 $\pi$ rad $$ 40 \degree \ \frac{2\pi \ {\rm rad}}{360 \degree} = 0.70 \ {\rm rad} $$
Know the three basic trigonometric functions

Gambar segitiga sikus-siku tegak: Sudut di kaki kiri ($\theta$), bidang miring (Hypotenuse), alas (Leg adjacent to $\theta$), tinggi (Leg opposite $\theta$)

$$ \sin \theta = \frac{\rm leg \ opposite \ \theta}{\rm hypotenuse} $$

$$ \cos \theta = \frac{\rm leg \ adjacent \ \theta}{\rm hypotenuse} $$

$$ \tan \theta = \frac{\rm leg \ opposite \ \theta}{\rm leg \ adjacent \ \theta} $$

Figure (3-11)