What Are Vectors?
In mathematics, a vector is a quantity that has both magnitudes and direction, but it doesn't display position. Examples of vector quantities: Velocity and acceleration. Vectors are shown by line segments, and the length of the line segment is the magnitude of the vector. For example, line AB is 7 cm, and it's pointed towards the south, hence, here AB is a vector. Boldface letters usually indicate vectors, like v. |v | represents the length and magnitude of a vector.
If a vector is multiplied by a scalar quantity, then the length of the vector is changed but not the magnitude. However, in the case of a negative number, the direction of the arrow will be reversed. In this article, we would learn about vectors joining two points.
Triangle Law for Vector Addition
The triangle law of vector addition states that, if two vectors are added, then the direction and the magnitude are shown at the two sides of the triangle. The third side thus represents the result of both the vectors.
If there are two vectors that are to be added, the first vector will be drawn as per the given scale. From the head of the first vector, the second vector will be drawn. Hence, the tail of the second vector will lie at the head of the first vector. Later, the third vector which will join the head of the first vector and the head of the second vector will be the required result.
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Equation of Vectors Joining Two Points
In the following example, points can be represented on x, y, and z-axes, respectively. Now if the two points are represented on x-, y- and z- coordinates, then it will be given as:
Point 1 as P1 (x1,y1,z1)
Point 2 as P2 (x2,y2,z2)
A vector will join both the points and the naming will be done as P1 and P2.
The vectors are represented from the origin I, along with the x-, y- and z-axes as i, j, and k, respectively.
Later, we have to join the origin O to P1 with the vector OP1, and origins O to P2 with the vector OP2.
Hence, by triangle law, we get:
→ → →
OP1 + P1P2 = OP2
Or
→ → →
P1P2 = OP2 - OP1
Then,
\[\overrightarrow{P_{1}P_{2}}\] = (x2î+y2ĵ+z2ƙ) - (x1ȋ+y1ĵ+z1ƙ)
= (x2-x1)î+(y2-y1)ĵ+(z2-z1)ƙ
The above-given equation represents the vector P₁ and P₂, and the magnitude.
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Example of Vector Joining Two Points
Here are a few vectors joining two points problems:
Question 1
Find the vector and its magnitude, which joins the point A (4, 5, 6) to point B (10, 11, 12).
Solution: As the vector is directed from point A to point B, it is denoted by
\[\overrightarrow{AB}\]
Hence,
\[\overrightarrow{AB}\]= (10-4) î + (11-5) ĵ + (12-6)ƙ
= 6î+6ĵ+6ƙ
The magnitude:
\[\overrightarrow{AB}\]= \[\sqrt{6^{2}+6^{2}+6^{2}}\] = \[\sqrt{36+36+36}\] = 108
= 10.39.
Question 2
Find the vector joining the points P with coordinates (1, 2, 3) and Q with coordinates(6, 5, 4) directed to point Q from P.
Solution: As the vector is directed from point P to point Q, it will be denoted as
\[\overrightarrow{PQ}\]
Hence,
\[\overrightarrow{PQ}\]= (6-1)î +(5-2)ĵ +(4-3)ƙ
= 5î +3ĵ+ ƙ
Question 3
Find the vectors joining the points P(2,3,0)and Q(-1,-2,-4), directed from Point P to Point Q
Solution: Given,
P = (2,3,0) and Q = (-1,-2,-4)
As P is directed towards Q, the arrow will be in the forward direction. Hence,
\[\overrightarrow{PQ}\] = (-1,-2)î + (-2,-3)j + (-4,-0)k̂
= -3î -5ĵ -4k̂
The vector joining two points P and Q are:
\[\overrightarrow{PQ}\] = -3î -5-ĵ -4k̂.
Question 4
Find the vector joining two points A(1,2,6), B(7,9,10), directed from Point A to Point B, Also find the Magnitude.
Solution: Given, A = (1,2,6) B = (7,9,10)
Because the vector is directed from point A to B, the arrow will move in forward direction, and it will be shown as
\[\overrightarrow{AB}\]
Then,
\[\overrightarrow{AB}\] = (7−1) î +(9−2)ĵ +(10−6) k̂
= 6î +7ĵ +4k̂
The magnitude of \[\overrightarrow{AB}\] will be given as
∣ \[\overrightarrow{PQ}\] ∣= \[\sqrt{(6 \times 6)+ (7 \times 7)+ (4 \times 4)}\]
= \[\sqrt{36 + 49 + 16}\]
= \[\sqrt{101}\]
=10.04
FAQs on Vector Joining Two Points
1. Why are Vectors Essential?
As vectors interpret the numbers quickly, analytical geometry becomes easier to understand. Vector is also very useful in 3D geometry. In real life, the vector is used for air traffic control, to know the magnitude and the direction. Force and velocity is the best example for vectors because both represent a particular direction. It's quite easy to find the vector equation of any line through given points; one can just take the sum of one of the points and a variable, thus scale the direction vector.
2. How to Find the Vector Equation of the Given Line with Two Points?
As a line is formed of a position and a direction, hence there will be two vectors. Position can be at any point in the line. The direction will always move from one point to the other. If the arrows go in the forward direction, then it will be represented by a positive sign, whereas if the arrow moves to the backward decision, then it will be represented by a negative sign. There are usually two points, an initial point, and a terminal point. One needs to join point A and point B through the origin with the help of the triangle law.
3. Who developed the concept of Vector?
The notion of vector, as we know it now, developed slowly over a period of more than 200 years. About a dozen persons made substantial inputs to its advancement. In 1835, Giusto Bellavitis generalized the underlying notion when he invented the concept of equipollence. Fundamentally, he developed an equivalence relation on the pairs of points in the plane, and so created the first set of vectors in the plane. The term vector was invented by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a Real number s (also called scalar) and a 3-dimensional vector. Like Bellavitis, Hamilton considered vectors as representational classes of equipollent directed segments. As complex numbers employ an imaginary unit to supplement the real line, Hamilton believed the vector v to be the imaginary portion of a quaternion.
4. What is vector calculus?
The term vector calculus is occasionally used as a substitute for the wider topic of multivariable calculus, which encompasses vector calculus as well as partial differentiation and multiple integration. Vector calculus serves a significant role in differential geometry and in the understanding of partial differential equations. Vector calculus was evolved from quaternion analysis by J. Willard Gibbs and Oliver Heaviside at the end of the 19th century. In the standard form employing cross products, vector calculus does not extend to higher dimensions, while the alternative technique of geometric algebra which employs exterior products.
5. What is a vector bundle?
In math, a vector bundle is a spatial development that helps make accurate the concept of a family of vector spaces parameterized by another space X. For instance, X can be a topological space, a manifold, or an algebraic variety, to every point x of the space X we associate a vector space V(x) in such a manner that these vector spaces fit together.
