Geometry of R 2 and R 3

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Geometry of R 2 and R 3 - PowerPoint PPT Presentation

Geometry of R 2 and R 3. Dot and Cross Products. Dot Product in R 2. Let u = (u 1 , u 2 ) and v = (v 1 , v 2 ) then the dot product or scalar product, denoted by u . v , is defined as u . v = u 1 v 1 + u 2 v 2. Dot Product in R 3.

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Geometry of R2 and R3

Dot and Cross Products

Dot Product in R2

Let u = (u1, u2) and v = (v1, v2) then the dot product or scalar product, denoted by u.v, is defined as

u.v = u1v1 + u2v2

Dot Product in R3

Let u = (u1, u2, u3) and v = (v1, v2, v3) then the dot product or scalar product, denoted by u.v, is defined as

u.v = u1v1 + u2v2 + u3v3

Example

Find the dot product of each pair of vectors

• u = (-3, 2, -1); v = (-4, -3, 0)
• u = (-4, 0, -2); v = (-3, -7, 6)
• u = (-6, 3); v = (5, -8)
Theorem 1.2.1

Let u and v be vectors in R2 or R3, and let c be a scalar. Then

• u.v = v.u
• c(u.v) = (cu).v = u. (cv)
• u.(v + w) = u.v + u.w
• u.0 = 0
• u.u = ||u||2
Theorem 1.2.2

Let u and v be vectors in R2 or R3, and let  be the angle they form. Then

u.v = ||v|| ||u|| cos

If u and v are nonzero vectors, then

Example

Find the angle between each pair of vectors.

• u = (-1, 2, 3); v = (2, 0, 4)
• u = (1, 0, 1); v = (-1, -1, 0)
Orthogonal Vectors

Two vectors u and v in R2 or R3 are orthogonal if u.v = 0.

Orthogonal, Normal, and Perpendicular, all mean the same.

Theorem 1.2.3

Let u and v be nonzero vectors in R2 or R3 and let  be the angle they form. Then  is

• An acute angle if u.v > 0
• A right angle if u.v = 0
• An obtuse angle if u.v < 0
Cross Product (Only in R3 )

Let u = (u1, u2, u3) and v = (v1, v2, v3) be nonzero vectors in R3. Then the cross product, denoted by u x v, is the vector

(u2v3 – u3v2, u3v1 – u1v3, u1v2 – u2v1)

Cross Product (Convenient notation)

Let u = (u1, u2, u3) and v = (v1, v2, v3) be nonzero vectors in R3. Then u x v, is the vector obtained by evaluating the determinant:

Example

Find the cross product of the following vectors

u = (-1, 1, 0); v = (2, 3, -1)

Theorem 1.2.4

The vector uxv is orthogonal to both u and v.

Theorem 1.2.4

Let u, v, and w be vectors in R3, and let c be a scalar. Then

• u x v = –(v x u)
• u x (v + w) = (u x v) + (u x w)
• (u + v)x w = (u x w) + (v x w)
• c(u x v ) = (cu)x v = u x (cv)
• u x 0 = 0 x u = 0
• u x u = 0
• ||u x v|| = ||u|| ||v|| sin  = (||u|| ||v|| – ||u.v||2)
Cross Product: Area

Let u, and v, be vectors in R3, Then the area of the parallelogram determined by u and v is

||u x v|| = ||u|| ||v|| sin 

Example

Find the area of the parallelogram determined by the vectors u = (-1, 1, 0) and v = (2, 3, -1).