The points B, C, and D are drawn on line segment A E...

The points $B, C,$ and $D$ are drawn on line segment $A E$ dividing it into four equal lengths. If $\overrightarrow{A D}=\vec{a},$ write each of the following in terms of $\vec{a}$ and $\vec{a}$. $$\begin{array}{lllll}\text { a. } \overrightarrow{E C} & \text { b. } \overrightarrow{B C} & \text { c. }|\overrightarrow{E D}| & \text { d. }|\overrightarrow{A C}| & \text { e. } \overrightarrow{A E}\end{array}$$

The points $B, C,$ and $D$ are drawn on line segment $A E$ dividing it into four equal lengths. If $\overrightarrow{A D}=\vec{a},$ write each of the following in terms of $\vec{a}$ and $\vec{a}$.
$$\begin{array}{lllll}\text { a. } \overrightarrow{E C} & \text { b. } \overrightarrow{B C} & \text { c. }|\overrightarrow{E D}| & \text { d. }|\overrightarrow{A C}| & \text { e. } \overrightarrow{A E}\end{array}$$

Key Concepts

Vector Representation

This concept involves representing points as position vectors and directed line segments as vectors in a coordinate system. A vector conveys both magnitude and direction, and it can be used to describe the displacement from one point to another, encapsulating the geometric layout in algebraic terms.

Vector Addition and Subtraction

Vector addition and subtraction allow for combining or comparing vectors. Addition corresponds to the sequential displacement from one point to another, while subtraction effectively finds the difference between two points. These operations are fundamental when expressing one segment in terms of another by adding or subtracting their corresponding vectors.

Scalar Multiplication

Scalar multiplication scales a vector by a numerical factor, affecting its magnitude but not its direction. This operation is essential when partitioning a line segment into proportional parts, such as dividing a segment into equal or specified ratios, by multiplying the vector by fractions that represent these divisions.

Internal Division of Line Segments

Internal division refers to the process of dividing a line segment into parts according to a given ratio. This concept is applied in vector geometry to determine the coordinates or vectors corresponding to points that partition a segment, facilitating the expression of new vectors as fractional sums of a given base vector.

Magnitude of a Vector

The magnitude of a vector is a measure of its length, typically expressed by the absolute value notation. It is used to quantify the displacement represented by a vector, differentiating between the directional (vector) form and the scalar length (distance) between two points.

Transcript

00:01 In this problem, we have been given a diagram which is shown over here.

00:06 Now in this diagram, it is said that the line segment ae is divided into four equal lengths by the points b, c, and d.

00:14 Now, the vector ad is equal to a, and we need to use that to write certain vectors in terms of b.

00:21 Now, in order to do that, first of all, note that the line segment ae is divided into four equal lengths by b, c, and d.

00:30 So that means that the length of ab is equal to the length of b c, that is equal to the length of cd, and that is equal to the length of d.

00:37 So that will mean that ad is divided into three equal lengths of a, b, c, and c, by these points b and c.

00:46 Now, since the entirety of ad is equal to the vector a, hence the vector a, b, will be equal to the vector b c, will be equal to the vector cd, and this will also be equal to the vector d .e because the length of d .e is the same and the direction is also the same and this will be one -third of a.

01:08 This is because a -b is one -third of the length of a -d.

01:13 Similarly, v -c is one -third of the length of a -d and so on.

01:17 So from here we can determine the given vectors.

01:21 The first one is the vector e -c.

01:24 We need to determine the vector e -c.

01:26 Now, ec can be written as ed plus dc.

01:33 Now, ed can be written as minus de.

01:37 Dc can be written as minus cd.

01:40 And from above, we can see that both de and cd is equal to one -third a.

01:44 So this will be minus one -third a, minus one -third a.

01:48 So this will be equal to minus 2 by 3 of a...