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The Parallelogram Law states that $$ {\mid a + b \mid}^2 + {\mid a - b \mid}^2 = 2 {\mid a \mid}^2 + 2 {\mid b \mid}^2 $$(a) Give a geometric interpretation of the Parallelogram Law.(b) Prove the Parallelogram Law. (See the hint in Exercise 62.)

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a) The sum of the square of the length of the diagonals of a parallelogram, are equal to the square of the length of the four sides of the parallelogram.b) $|\mathbf{a}+\mathbf{b}|^{2}+|\mathbf{a}-\mathbf{b}|^{2}=(\mathbf{a}+\mathbf{b}) \cdot(\mathbf{a}+\mathbf{b})+(\mathbf{a}+\mathbf{b}) \cdot(\mathbf{a}+\mathbf{b})$$=2(\mathbf{a} \cdot \mathbf{a})+2(\mathbf{b} \cdot \mathbf{b})$$=2|\mathbf{a}|^{2}+2|\mathbf{b}|^{2}$$=L . H . S$$=R . H . S \quad$ (R.T.P)

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 3

The Dot Product

Vectors

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

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The Parallelogram Law stat…

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Parallelogram Identity The…

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A parallelogram is formed …

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Two sides of a parallelogr…

The problem is the parallelogram law states that magnitude of a plus b, squared plus magnitude of a minus b squared is equal to 2 times magnitude of a square plus 2 times magnitude of v square part a gives a geometric interpretation of the parallelogram law. So, for part a if we draw a parallelogram if this is vector a vector, a vector b, vector band, the 2 delganos a and plus c. This is a plus t and a minus b. This is a minus b poi, so the parallelogram law states that the sum of the square of diagonals is equal to the sum of the square of 2. The 4 th size of this parallelogram proof that proved the parallelogram law. So first, we use magnitude of a plus b square is equal to a plus b delta. A plus b and the magnitude of a minus b squared is equal to a minus b dot a minus b. This is equal to a square, a dot, a plus a dot b, plus b dot, a plus e dot b, and this is equal to a dot, a minus, a dot, a minus, a dot, a plus e dot, so magnitude of a plus b square plus magnitude. Of a minus b squared so we can see this is equal to a dot, a multiplied 2 plus b dot b, multiplied 2. So this is equal to 2 times magnitude of a square plus 2 times the magnitude of the square.

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