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Vectors

Vectors are directed line segments in which we precisely know which point is the initial point and which one is the terminal point. You can make two different vectors from two different points in a plane. If you have point A and point B in a plane, you can make vector that goes from point A to point B and another vector that goes from point B to point A. If you remember, line segment that connects point A and B is marked with a line above the letters that mark initial and terminal point: \bar{AB}. Vectors will be marked similarly but with an arrow above the initial and the terminal point – \vec{AB} or \vec{BA}, with first letter marking the initial point and second the terminal point.

vector abvector ba

Vectors are uniquely determined by their magnitude, direction and orientation.

 

Drawing vectors in a plane

Every vector is determined by two points. This makes drawing them in a plane quite easy. For example, if we are given two points A and B where A = (1, 2) and B = (5, 6) and our task is to draw vector \vec{AB}, we would first draw a line segment \bar{AB} and then just draw an arrow which marks our terminal point.

vector in coordinate plane

To every point in a plane can be assigned a unique vector whose initial point is in origin and the terminal point is in the given point. These vectors are very important in vector geometry and they are called position vectors.

 

Magnitude of vectors

Magnitude of vector \vec{AB} is the length of a line segment \bar{AB}.
 

Some other names for vector magnitude: vector norm, vector modulus or absolute value of a vector.

Magnitude of any vector is determined by the placement of its initial and terminal point, and is calculated exactly the same as the length of a line segment. The magnitude of a vector is always a positive number or a zero. Magnitude of a vector is equal to zero if and only if the initial point is equal to the terminal point.

If you have two points X = (x_1, y_1) and Y = (x_2, y_2), magnitude of a vector \vec{AB} or is:

\mid \vec{AB} \mid = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Of course, if you have position vector, whose end point is X = (x, y)

\mid \vec{x} \mid = \sqrt{x^2 + y^2}

Vector whose magnitude is equal to 1 is called the unit vector, and vector whose magnitude is equal to zero is called the null vector.

Example 1. Draw vectors in a plane and calculate their magnitude.

1. \vec{AB} where A = (1, 5), B = (4, 2)

2. \vec{CD} where C = (0, 0), D = (5, - 5)

3. \vec{EF} where E = (6, 7), F = (6, 7)

1. \vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(4 - 1)^2 + (2 - 5)^2} = \sqrt{18}

\sqrt{18} = \sqrt{9 * 2} = 3\sqrt{2}

2. \vec{CD} = \sqrt{x^2 + y^2} = \sqrt{5^2 + (-5)^2} = \sqrt{50} = \sqrt{2 * 25} = 5\sqrt{2}

3. \vec{EF} = 0

two vectors in coordinate plane

 

Direction of vectors

The direction of a vector is the measure of the angle it encloses with the y – axis. It can be observed as the slope of the line it lies on, because of that its slope is calculated in the same way you calculate the slope of a line. We’ll mark the direction of a vector with φ.

If you have a vector \vec{PQ}, P = (x_1, y_1), Q = (x_2, y_2)

tan \alpha = \frac{y_2 - y_1}{x_2 - x_1}

Example 2. What is the direction of a vector a whose initial point is A = (1, 3), and terminal point B = (3, 5)?

Formula says: tan⁡ \alpha = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 3}{3 - 1} = 1 \rightarrow \alpha = 45^{\circ}.

directions of vector a given graphically

 

Orientation of vectors

Vector orientation is exclusively related to collinear vectors. Vectors are collinear if lines on which they lie are parallel. Let’s say you have two vectors \vec{AB} and \vec{CD}. Their orientation can only be equal or contrary.

How do we determine orientation? Let’s assume that those two vectors are collinear. First we take one vector and translate it in a way that its initial point falls into the initial point of other vector. That means that we translate vector \vec{CD} so that C = A Now, if the points B and D are on the same side of point A, they are of same orientation, but if they are on different sides they are of contrary orientation.

Vectors with the same orientation:

vectors with same orientationaddition of same orientation vectors

Vectors with contrary orientation:

two vectors with an opposite orientation

subtraction of two vectors

Note: Two vectors are equal if they have the same magnitude, direction and orientation. If two collinear vectors are of equal length, but different orientation they are called contrary vectors. If a the vector we’re observing, his contrary vector is marked with – a.

 

Vector addition

By adding vectors \vec{AB} and \vec{CD} we’ll get a new vector whose initial point is the initial point of the first addend and the terminal point the terminal point of the second addend. By applying this rule a triangle accrues. This is why this way of addition is called the triangle rule. When we have two vectors that we have to add together, first we have to translate one vector onto the other one, in a way that the terminal point of the first is the initial point of the second. Then, all we have left to do is to complete the triangle and mark the orientation of our new vector.

Step by step triangle rule addition \vec{AB} + \vec{AB}:

addition-of-vectors-using-triangle-rule addition-of-vectors-using-triangle-rule2 addition-of-vectors-using-triangle-rule3

Addition with null vector:

\vec{a} + \vec{0} = \vec{a}

\vec{0} + \vec{a} = \vec{a}

 

The parallelogram law of vector addition

If you have two vectors that have the same initial point, you have to use the parallelogram law of vector addition. You simply consider those two vectors as adjacent sides of a parallelogram, their sum will the diagonal of that parallelogram. The initial point of the sum of these two vectors will be their initial point.

Step by step:

addition-using-parallelogram-law-of-vectors addition-using-parallelogram-law-of-vectors2 addition-using-parallelogram-law-of-vectors3

Properties of addition

Addition of vectors is commutative:

\vec{a} + \vec{b} = \vec{b} + \vec{a}

Addition of vectors is associative:

(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})

 

Addition of more than two vectors

Addition of n inter-related vectors \vec{A_1 A_2}, \vec{A_2 A_3}, \vec{A_3 A_4}, ... , \vec{A_n-1 A_n} is equal to the vector \vec{A_1 A_n}

addition more than two vectors

Note: As you already know, -a is a vector whose orientation is contrary from orientation of vector a, but their magnitude and direction are equal. Using this statement we know how to subtract vectors. If ve have two vectors, a and b, and we need to subtract b from a, (a - b) we’d simply change the orientation of vector b ⃗ and add them like that - a + (-b).

 

Multiplication of vectors and real numbers

The product of real number k \not = 0 and vector \vec{a} is a vector, which we denote as k \vec{a}, with rules:

  • Vectors \vec{a} and k \vec{a} are collinear vectors of equal orientation if k > 0 and of contrary orientation if k < 0
  • Magnitude of vector k \vec{a} is equal to \mid k \mid * \mid \vec{a} \mid
  • The product of a vector and zero is a null-vector
  • The product of a vector and 1 is that vector

For example, if you have a vector \vec{AB}, where A = (2, 4), B = (5, 6), calculate 2 \vec{AB} and -2 \vec{AB}, the product of a vector and a real number will always lie on the line on which our observed vector lies. This is why the first thing we’ll do is to draw a line on which our vector \vec{AB} lies.

vector ab in coordinate plane

Now we want to calculate 2 \vec{AB}. It’s magnitude will be twice the magnitude of a vector \vec{AB}. That vector will lie on the same line and will have the same orientation because 2 > 0. Now we have all the data we need:

multiplication of two equal vectors

Now we want to calculate -2 \vec{AB}. Since -2 < 0 this vector will have contrary orientation from the orientation of vector \vec{AB}, double magnitude and the same direction.

calculation of ab vector with an opposite orientation

Properties of multiplying a vector with a real number

It is valid for every two real numbers k and l, and for every two vectors a and b:

k (\vec{a} + \vec{b}) = k \vec{a} + k \vec{b}

(k + l) \vec{a} = k \vec{a} + l \vec{a}

(kl) \vec{a} = k (l \vec{a})

 

Linear combination – linear dependence and independence

If \vec{a} and \vec{a} are vectors and α and β real numbers, vector

\vec{c} = \alpha \vec{a} + \beta \vec{b}

Is called the linear combination of vectors \vec{a} and \vec{b} with coefficients \alpha and \beta

If \vec{a} and \vec{b} are two collinear not null-vectors with the same orientation where \vec{a} is k times longer than \vec{b}, then we can write:

two collinear vectors same orientation

This means that one vector can be represented using the other. This means that \vec{a} and \vec{b} are linearly dependent.

The vectors \vec{a_1}, \vec{a_2} ... , \vec{a_n} are said to be linearly dependent if there exist real numbers \alpha_1, \alpha_2, ... , \alpha_n such that:

\vec{0} = \alpha_1 \vec{a_1} + \alpha_2 \vec{a_2}, ... , \alpha_n \vec{a_n}

If linear combination \alpha_1 \vec{a_1} + \alpha_2 \vec{a_2}, ... , \alpha_n \vec{a_n} is equal to \vec{0} only when \alpha_1, \alpha_2, ... , \alpha_n are all equal to zero, then it is said that vectors \vec{a_1}, \vec{a_2} ... , \vec{a_n} are linearly independent.

Every two collinear vectors in a plain are linearly dependent and every two non-collinear vectors are linearly independent.

Every vector in a plain can be presented in a unique way as a linear combination of two non-collinear vectors.

 

Vectors in a coordinate plane

Vectors \vec{i} and \vec{j}
If E is the unit point on the x-axis and F is the unit point on y-axis, vector OE is marked with \vec{i} and vector OF with \vec{j}

vectors i and j

Now you may wonder what is the purpose of these two vectors is. Using these two vectors we can present any vector in a plain in a unique way.

If point P has coordinates (x, y) then the position vector \vec{OT} has presentation \vec{OF} = x \vec{i} + y \vec{j}

If A_1 = (x_1, y_1) and A_2 = (x_2, y_2) are two points of plane then: \vec{A_1, A_2} = (x_2 - x_1) \vec{i} + (y_2 - y_1) \vec{j}

For example if we have two points A = (1, 3) and B = (2 , 5) and we have to find vector \vec{AB} :

\vec{AB} = (x_2 - x_1 ) \vec{i} + (y_2 - y_1) \vec{j} = (2 - 4) \vec{i} + (5 - 3) \vec{j} = \vec{i} + 2 \vec{j}

The length of a vector:

If \vec{a} = x \vec{i} + y \vec{j} \Rightarrow \mid \vec{a} \mid = \sqrt{x^2 + y^2}

If \vec{a} \not= \vec{0} is given vector. The unit vector of vector \vec{a} is a vector:

unit of a vector

 

Example: find the unit vector of a vector \vec{a} = 4 \vec{i} - 2 \vec{j}

\mid \vec{a} \mid = \sqrt{4^2 + (-2)^2} = \sqrt{20}

\vec{a_0} = \frac{1}{\mid \vec{a} \mid} \vec{a} = \frac{4 \vec{i} - 2 \vec{j}}{\sqrt{20}} = \frac{4 \vec{i}}{\sqrt{20}} - \frac{2 \vec{j}}{\sqrt{20}}

 

Scalar or dot product

Let’s say there are two vectors in a plain, both different from vector 0. If they don’t have the same initial point, we simply translate one vector onto the other in a way that they do have the same initial point.
The angle between those two vectors is the smaller angle of the two angles enclosed by the half lines those two vectors are lying on.

scalar or dot product

If \vec{a} and \vec{b} are two vectors different from \vec{0}, the product

\mid \vec{a} \mid * \mid \vec{b} \mid cos \angle (\vec{a}, \vec{b})

Is called the scalar or dot product of vectors \vec{a} and \vec{b}.

If any of these two vectors is a null-vector we define that:

\vec{0} * \vec{a} = 0

The result of scalar product is real number.

If \vec{a} and \vec{b} are perpendicular, their dot product is equal to zero because in the definition of scalar product we see that we have to multiply with the cosine of the angle between them, if they are perpendicular that angle is equal to 90^{\circ} and cos 90^{\circ} = 0.

If \vec{a} = a_1 \vec{i} + a_2 \vec{j} and \vec{b} = b_1 \vec{i} + b_2 \vec{j}

\vec{a} * \vec{b} = a_1 b_1 + a_2 b_2

Example. If \vec{a} = 2 \vec{i} + 3 \vec{i} and \vec{b} = \vec{i} - 5 \vec{i} calculate \vec{a} * \vec{b}.

\vec{a} * \vec{b} = 5 * 2 + 3 * (- 5) = - 5

Example. If \vec{a} = 2 \vec{i} and \vec{b} = \vec{i} - 2 \vec{i} calculate \vec{a} * \vec{b}.

\vec{a} * \vec{b} = 2 * 1 + 0 * (- 2) = 2

scalar vectors

Properties of scalar products

1. (\alpha * \vec{a}) * \vec{b} = \alpha * (\vec{a} * \vec{b})

2. (\vec{a} + \vec{b}) * \vec{c} = \vec{a} \vec{c} + \vec{b} \vec{c}

3. \vec{a} * \vec{a} \ge 0

4. \vec{a} * \vec{a} = 0 \leftrightarrow \vec{a} = \vec{0}

\vec{a}, \vec{b}, \vec{ac} ; a \epsilon R.

Worksheets

  Naming vertexes and vectors (419.4 KiB, 129 hits)

  Vectors measurement of angles (490.3 KiB, 171 hits)

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