[b][size=200][size=150]What is a Vector?[/size][/size][br][/b][br]Vectors are often used in physics to represent quantities like force and velocity when we need to specify both the magnitude and direction of those quantities. Usually, we represent a vector visually by an arrow. In the applet below, vectors in [math]\mathbb{R}^2[/math] and [math]\mathbb{R}^3[/math] are shown. [br][br][b]Remark[/b]: There is a very special vector that does not have any direction. What is it?

Vectors can be drawn as arrows pointing out from the origin (the point where all axes intersect). On a coordinate space (either 2D or 3D), the coordinates of the point at the arrowhead uniquely determine the vector. Therefore, for the vectors [math]\vec{v}[/math] in [math]\mathbb{R}^2[/math] and [math]\vec{u}[/math] in [math]\mathbb{R}^3[/math], we can express them mathematically as follows:[br][br][math]\vec{v} = \langle v_1 , v_2 \rangle[/math] and [math]\vec{u}= \langle u_1, u_2, u_3 \rangle[/math][br][br]where [math](v_1 , v_2)[/math] and [math](u_1, u_2, u_3)[/math] are points at the arrowhead of [math]\vec{v}[/math] and [math]\vec{u}[/math] i.e. P and Q respectively. [br][br][math]v_1[/math] and [math]v_2[/math] are called the [b]x-component[/b] and [b]y-component[/b] of [math]\vec{v}[/math] respectively.[br][br]Similarly, [math]u_1[/math], [math]u_2[/math], and [math]u_3[/math] are called the [b]x-component[/b], [b]y-component[/b], and [b]z-component[/b] of [math]\vec{u}[/math] respectively.[br]

(You can freely drag the points P and Q on the arrowheads of both vectors in the applet above to change the length and the direction of the vectors.)[br][br]Moreover, two vectors are regarded as equal if they have the same length and direction. Hence, the tail of a vector can be any point other than the origin i.e. we may shift the vector to another position if necessary, as long as the length and direction remain unchanged. Therefore, whenever you are given a vector in [math]\mathbb{R}^2[/math] or [math]\mathbb{R}^3[/math] and want to express it mathematically, you can just shift the vector to the origin and find the coordinates of the arrowhead.[br][br](You can freely drag the green vectors in the applet above to any position you like and they are considered as the same vectors as the black ones pointing out from the origin.)

[size=150][u][size=100]Postion Vectors[/size][/u][/size][br][br]As mentioned before, vectors are uniquely determined by the points at their arrowheads when pointing from the origin. Therefore, any point P (or Q) in [math]\mathbb{R}^2[/math] (or [math]\mathbb{R}^3[/math]) can be represented by the vector with its tail at origin and its head at the point P (or Q). It is called the [b]position vector[/b] of P (or Q). From this perspective, points and vectors are just two sides of the same coin. Sometimes it is convenient to regard points as position vectors.[br][br]

[u]Norm of a Vector[/u][br][br]How can we compute the magnitude (length) of a vector? Thanks to Pythoragas Theorem, we have the following definitions:[br][br]Given [math]\vec{v}=\langle v_1,v_2\rangle[/math] in [math]\mathbb{R}^2[/math], the [b]norm[/b] of the vector [math]\vec{v}[/math] is [br][br][math]|\vec{v}|=\sqrt{v_1^2+v_2^2}[/math][br][br]which is exactly the magnitude (length) of the vector [math]\vec{v}[/math].[br][br]For [math]\vec{u}=\langle u_1,u_2, u_3\rangle[/math] in [math]\mathbb{R}^3[/math], the [b]norm[/b] of the vector [math]\vec{u}[/math] is [br][br][math]|\vec{u}|=\sqrt{u_1^2+u_2^2+u_3^2}[/math][br][br]which is exactly the magnitude (length) of the vector [math]\vec{u}[/math].[br][br]By definition, the norm of zero vector is zero and the norm of any non-zero vector is positive.[br][br][br]The following applet show how the above formulas are derived from Pythoragas theorem.[br][br][br]

[b][size=150]Vector Addition[/size][/b][br][br]There are two main operations on vectors. The first one is the addition of two vectors. First, we consider two vectors in the plane, we can define their addition visually using the applet below:[br][br][list=1][*]Construct two vectors u and v in [math]\mathbb{R}^2[/math] using the vector tool [icon]/images/ggb/toolbar/mode_vector.png[/icon].[/*][*]Drag u to the origin i.e. the tail of u is a the origin.[/*][*]Drag v to the arrowhead of u.[/*][*]The vector u + v is defined as the vector pointing from the origin to the arrowhead of v.[/*][/list][br][br]This is the so-called [b]triangle rule[/b].[br][br]([i]Note[/i]: Here we write "u" and "v" instead of [math]\vec{u}[/math] and [math]\vec{v}[/math] for the sake of convenience.)

Alternatively, you can regard the vector u + v as the "diagonal vector" of the parallelogram formed by the two vectors u and v pointing out from the origin. This is the so-called [b]parallelogram rule[/b]. For the physics viewpoint, this definition of addition is quite natural. You can imagine two forces represented by u and v act on a mass at the origin. The resultant force is exactly u + v.[br][br]In the left pane of the above applet, when a vector u is created, its components [math]u=\langle u_1, u_2 \rangle[/math] are shown in the following format: [math]u=\begin{pmatrix} u_1 \\ u_2 \end{pmatrix}[/math]. Observe the components of the vectors when you add two vectors together and answer the following question:[br]

The addition of two vectors in [math]\mathbb{R}^3[/math] are defined in the same way i.e. either using triangle rule or parallelogram rule. Moreover, suppose we are given two vectors [math]\vec{u}=\langle u_1,u_2,u_3 \rangle[/math] and [math]\vec{v}=\langle v_1,v_2,v_3 \rangle[/math] in [math]\mathbb{R}^3[/math],[br]

[size=150][b]Vector Scaling[/b][/size][br][br]The second main operation on vectors is scaling. Suppose k is any real number and u be any vector in [math]\mathbb{R}^2[/math] or [math]\mathbb{R}^3[/math].[br][list][*]If k >0, then ku is the vector having the same direction as u such that its length is k times the length of u.[br][/*][*]If k = 0, then ku is a zero vector.[/*][*]If k < 0, then ku is the vector having the opposite direction to u such that its length is |k| times the length of u. (Note: |k| is the absolute value of k.). In other words, [math] |ku|=|k||u|[/math]. [/*][/list][br]In the applet below, you first construct a "slider" corresponding to the value of k by typing "k" and press Enter. Then construct a vector u and its scaling by k i.e. the vector ku. You can drag the slider to see how the vector ku changes for different values of k.[br]

[u]Unit Vectors[br][/u][br]A vector is called a [b]unit vector[/b] if its norm (length) equals 1. For any non-zero vector [math]\vec{v}[/math], we can scale it by factor [math] \frac{1}{|\vec{v}|}[/math] to unit vector in the same direction:[br][br]Let [math]\vec{u}=\frac{1}{|\vec{v}|} \vec{v}[/math]. Then [math]|\vec{u}|=\left|\frac{1}{|\vec{v}|} \vec{v}\right|=\frac{1}{|\vec{v}|}|\vec{v}|=1[/math].[br][br]In [math]\mathbb{R}^2[/math], there are two special unit vectors [math]\vec{i}=\langle 1,0\rangle[/math] and [math]\vec{j}=\langle 0,1\rangle[/math] such that any vector [math]\vec{v}[/math] in [math]\mathbb{R}^2[/math] can be expressed in terms of [math]\vec{i}[/math] and [math]\vec{j}[/math] as follows:[br][br][math]\vec{v}=\langle v_1,v_2\rangle =v_1\langle 1,0\rangle+v_2\langle 0,1\rangle= v_1\vec{i}+v_2\vec{j}[/math].[br][br]Similarly, in [math]\mathbb{R}^3[/math], the three special unit vectors are [math]\vec{i}=\langle 1,0,0\rangle[/math], [math]\vec{j}=\langle 0,1,0\rangle[/math], and [math]\vec{k}=\langle 0,0,1\rangle[/math]. Moreover, any vector [math]\vec{u}[/math] in [math]\mathbb{R}^3[/math] can be expressed as follows:[br][br][math]\vec{u}=\langle u_1,u_2,u_3\rangle =u_1\langle 1,0,0\rangle+u_2\langle 0,1,0\rangle+u_3\langle 0,0,1\rangle= u_1\vec{i}+u_2\vec{j}+u_3\vec{k}[/math].[br][br][br][br][br]

[u]Vector Subtraction[/u][br][br]Vector subtraction can be easily defined in terms of addition and scaling as follows: [math]\vec{u} -\vec{v} = \vec{u} + (-1)\vec{v}[/math]. Also, the components of [math]\vec{u}-\vec{v}[/math] can be computed by doing the subtraction component-wise.[br][br]You can construct vectors [math]\vec{u}[/math] and [math]\vec{v}[/math] in the above applet and then find out [math]\vec{u} + (-1)\vec{v}[/math].[br][br]

Suppose [math]A=(a_1,a_2)[/math] and [math]B=(b_1,b_2)[/math] are two points in [math]\mathbb{R}^2[/math]. The vector with [math]A[/math] as its tail and [math]B[/math] as its arrowhead is denoted by [math]\overrightarrow{AB}[/math]. As we know, [math]\langle a_1,a_2\rangle[/math] and [math]\langle b_1,b_2\rangle[/math] are the position vectors of [math]A[/math] and [math]B[/math] respectively. Therefore, we have[br][br][math]\overrightarrow{AB}=\langle b_1,b_2\rangle-\langle a_1,a_2\rangle=\langle b_1-a_1,b_2-a_2\rangle[/math][br][br]Since the norm of [math]\overrightarrow{AB}[/math] is exactly the distance [math]d[/math] between [math]A[/math] and [math]B[/math], we have[br][br][math]d=|\overrightarrow{AB}|=\sqrt{(b_1-a_1)^2+(b_2-a_2)^2}[/math], we have[br][br][br]Similarly, for any two points [math]P=(p_1,p_2,p_3)[/math] and [math]Q=(q_1,q_2,q_3)[/math] in [math]\mathbb{R}^3[/math], [br][br][math]\overrightarrow{PQ}=\langle q_1-p_1,q_2-p_2,q_3-p_3\rangle[/math][br][br]Moreover, the distance [math]d[/math] between [math]P[/math] and [math]Q[/math] is as follows:[br][br][math]d=|\overrightarrow{PQ}|=\sqrt{(q_1-p_1)^2+(q_2-p_2)^2+(q_3-p_3)^2}[/math][br][br][br][br][br][br] [br][br]