Parallelogram law of vector addition class 11

Two or more vectors can be added to give another vector, which is called the resultant of the vectors. The resultant would produce the same effect as that of the original vectors together. (I)Geometrical method: To find + , shift such that its initial point coincides with the terminal point of . Now, the vector whose initial point coincides with the initial point of , and terminal point coincides with the terminal point of represents ( + ) as shown in the above figure. To find ( + ), shift such that its initial point coincides with the terminal point of . A vector whose initial point coincides with the initial point of and terminal point coincides with the terminal point of represents ( + ). Illustration 1. If the position vector of point A and B are and respectively. Find the position vector of middle point of AB. Solution: (II)Parallelogram law of addition of vectors: If two vectors are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, their resultant will be represented in magnitude and direction by the diagonal of the parallelogram drawn from that point. We get , R 2 = P 2 + Q 2 + 2PQ cosθ φ = tan-1 where φ is the angle that the resultant makes with (i) θ = 0 0 and are in the same direction i.e. they are parallel cos 0 0 = 1 ∴ | | = | | + | | &φ = 0 0 (ii) θ = 180 0, and are in opposite direction i.e. they are antiparallel cos180 0 = -1 ∴ | | = | | ~ | | and is in the direction of the larger vector. (iii) θ = 90 o, ...

A Parallelogram is a four-sided quadrilateral whose opposite sides are parallel and congruent to each other. The opposite angles of a parallelogram are equal. The parallelogram and a rectangle are near about the same with one distinguishing property that the rectangle has all the angles of 90° 0 and that of parallelogram does not. In Mathematics, the parallelogram law belongs to elementary Geometry. This law is also known as parallelogram identity. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. In this article, let us study the definition of a parallelogram law, proof, and parallelogram law of Parallelogram Law of Addition Parallelogram law of addition states that the sum of the squares of the length of the four sides of a parallelogram equals the sum of the squares of the length of the two diagonals. In Euclidean geometry, it is a must that the parallelogram should have equal opposite sides. (Image to be added soon) If ABCD is a parallelogram, then AD = BC and AB = DC. Then according to the definition of the parallelogram law, it is stated as 2(AB) 2 + 2(BC) 2 = (AC) 2 + (BD) 2 . If a parallelogram is a rectangle, then the law is stated as 2(AB) 2 + 2(BC) 2 = 2(AC) 2 Because in a rectangle, two diagonals are of equal lengths. i.e., (AC=BD) Parallelogram Law of Vectors If two vectors say vector p and vector q are acting simultaneously at a point, then it can be rep...

Let Pand Qbe two vectors acting simultaneously at a point and represented both in magnitude and direction by two adjacent sides OA and OD of a parallelogram OABD as shown in figure. Letθ be the angle betweenP andQ andR be the resultant vector.Then, according to parallelogramlaw of vector addition, diagonalOB represents the resultant of Pand Q. So, we have R = P + Q Now, expand A to C and draw BC perpendicular toOC. From triangle OCB, OB 2 = OC 2 + BC 2 Or, OB 2 = (OA + AC) 2 + BC 2 ......(i) In triangle ABC, cos \(\theta\)= \(\frac\) or, BC = AB sin \(\theta\) or, BC = OD sin \(\theta\)= Q sin \(\theta\)[ \(\because\)AB = OD = Q] Substituting value of AC and BC in (i), we get which is the magnitude of resultant. Let \(\phi\)be the angle made byresultantR withP. Then, From triangle OBC, which is the direction of resultant. Special cases: (a) When two vectors are acting in same direction: Thus, the magnitude of the resultant vector is equal to the sum of the magnitude of the two vectors acting in the same direction and their resultant acts in the direction of P and Q. (b) When two vectors act in the opposite directions: Thus, the magnitude of the resultant of two vectors acting in the opposite direction to the difference of the magnitude of two vectors and it acts in the direction of bigger vectors. (c) If two vectors act perpendicular to each other: Categories • • (31.9k) • (8.8k) • (764k) • (261k) • (257k) • (218k) • (248k) • (2.9k) • (5.2k) • (664) • (121k) • (72.1k) • (3...

Vector, in physics, a quantity which has both magnitude as well as direction. It is usually denoted by an arrow whose direction is the same as that of the given quantity and length is proportional to the magnitude of the quantity. Although a vector has magnitude and direction, it has no position. That is, as long as its length does not change, a vector does not change as it moves parallel to itself. Vector Analysis Vector analysis is a branch of mathematics which is concerned with quantities that have magnitude and direction. Some physical and geometrical quantities, known as scalar quantities, can be defined by expressing their magnitude in appropriate units of measurement. Therefore, mass is defined in grams, temperature is expressed in degrees on a scale, and time is expressed in seconds. Scalars can be represented graphically by points on a numeric scale, like a clock or thermometer. There are also some quantities, termed as vectors, which require an indication of direction and magnitude. Vector Vector is a mathematical structure. It has many uses in physics as well as in geometry. We know that the position of points on the coordinate plane can be expressed by the ordered pair as (x, y). Using vectors is very useful in simplifying three-dimensional geometry. When we measure a physical quantity, the measured value we get is always a number. This number only makes sense if the corresponding unit is specified. The result of a measurement therefore has a numerical value an...