Calculus III: Two Dimensional Vectors (Level 1 of 13) | Basics

Two-Dimensional Vectors (Level 1) In the next series of videos we are going to around vectors cover both geometric and algebraic. In general, carriers were used to represent lines and levels in geometry, and was also used to represent quantities such as force and speed in physics. Many quantities in geometry and physics, such as as length, area, volume, temperature, mass, and time, can be described by a single true number scaled down to a suitable unit of measurement such as grams, which is a unit of mass. These quantities are called scalar quantities, and the actual number associated with with each quantity becomes a scalar. Quantities, such as power, displacement, velocity, and acceleration, involves both a magnitude and a direction and can not be characterized completely by a single real number. a convenient way of consuming these types of quantities is by the use of vectors. A vector is essentially a directed line segment represented by a (Arrow). The directing line segment is constructed by an initial point P also known as (The tail) and an endpoint Q also known as (the tip or head) of the vector.

We can indicate this vector by writing the letters of the initial point in this case P and the terminal point in this case Q, and then we draw an arrow on top of the letters as follows. Similar to a line segment from geometry, vectors a definite measurable length, the length of a vector is referred to as the size of the vector and is indicated as follows, using the absolute value symbol or by using double vertical bars as follows, in these videos I will use the double vertical bars to indicate the size of a vector, some textbooks will use the absolute value symbol, unfortunately, this notation can be misleading, many students think they need the "absolute find value" rather than the size of a vector. Because of this potential confusion, I will use the double vertical bars to indicate the scope of a vector. Next let's talk about equivalent vectors.

In general, directed line segments that have the same length (Magnitude) and direction It is said to be equivalent. For example, the following 2 vectors equal because the length or extent of each vector is the same and they are both shown in the same direction. On the other hand the following two vectors are not equivalent vectors because one vector is longer than the second vector, in other words the first vector has a larger size compared to the second vector, notice that both vectors show in the same direction, although they share this common feature that they do not consider be equal vectors because both the magnitude and direction need to be the same.

Just like the next two vectors are also not equal, see that they are the same length but each vector points in a different direction, in this scenario only one of the two requirements for vector equivalence is satisfied. as a final, for example the following two vectors not equivalent since both vectors have distinct sizes and directions, in this case no of the two requirements for vector equivalence factors is satisfied. Keep in mind that both the size and direction need to be considered the same for two or so vectors equally. Another notation used to indicate vectors is by assigning a small, bold letter like u, v and w, in practice it is quite difficult to write bold letters on paper, such an alternative way of indicating vectors is by using a lower case letter with an arrow above similar to the way we indicated rays in geometry class. In this way we can indicate vector PQ as vector v.

If we wanted to indicate that two vectors are equal or (equal) all we have to do is place an equal sign between the two symbols as follows. vector v equal vector u. Finally, we can indicate a special vector that has no specific direction this vector is known as the zero vector and is indicated as follows, this vector has a length of zero hence the zero vector it is usually represented as a single point you can think of of this vector as with the same initial and endpoint. The direction of zero vectors can technically point in any direction but we will not assign the effort of a specific direction. Let's go over some examples. Name all the equivalent vectors in the parallelogram showed. Remember that two vectors are equal if they are the same direction and size but they do not necessarily have to be at the same place. Since we are told that this is a parallelogram both pairs opposite sides congruent and parallel, also bisecting the diagonals of the parallelogram other.

With this information at hand we can start naming all the equivalent vectors. Remember that vectors are direction sensitive so we need to indicate the initial or tail first result by the main or end point. Vector AB is equivalent to vector DC since both vectors has the same length; they are parallel and both point in the same direction. In the same wise vector AD and vector BC are also similar as they are the same size and direction. Let's take a look at the diagonals, we know from elementary geometry that the diagonals of a parallelogram bisect each other this means that vector DE and vector EB have the the same size or length, they also lie together the same line and point in the same direction, see that all the requirements for vector equivalence factors are met so these carriers are equivalent.

Finally, vector AE and vector EG are also equivalent because they have the same size and direction, again, due to the geometric properties of parallelograms. Let go well about the final example. Name all the equivalent vectors shown in the figure. Quadrilateral ABCD is congruent with quadrilateral CEGF. Okay, remember that when two digits are congruent corresponding angles and corresponding sides of the figure have the same measurement. This means that all of these carriers have the same length or size. Also, because right forms vertical angles at point C the corresponding sides of the quadrilaterals are also parallel to each other.

With this information we can deduce that vector BA is equivalent to vector GF, Vector CB is equivalent vector GA, Vector DC is equivalent to vector CE and Vector AD is equivalent to vector CF. Keep in mind that both the size and direction of the vectors must be the same. Ok, in our next video we will start learning how to describe vectors in a two-dimensional coordinate system..

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