phy10-unit vector.pdf
TRANSCRIPT
UNIT VECTORS- are vectors whose magnitude is exactly one(1) and used to point a
particular direction.
- points to (+)x – axis.
- points to (+)y – axis.
- points to (+)z – axis.x
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UNIT VECTORS- are used as a another way of writing a vector.
- Vector Components
- Scalar Components
- Magnitude of the Vector
A = Ax i + Ay j + Az ki
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Ax ii
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Ay ji
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Az ki
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Ax
Ay
Az
= = = =A = Ax2 + Ay2 + Az2
y
x53.13o
UNIT VECTORS- are used as a another way of writing a vector.
A = Ax i + Ay j + Az ki
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k
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kAx = 5 Cos 53.13o
Ax = 3
Ay = 5 Sin 53.13o
Ay = 4
Az = 0AAy ji
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kAx ii
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A = 5 v, 53.13o N of E = (3 i + 4 j ) vi
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Given the vector A = 5 v, 53.13o N of E;
Ay j = -2 ji
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UNIT VECTORS
A vector, A = 4 i - 2 j + 3 k , would mean;
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ji
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Ax i = 4 ii
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kAz k = 3 ki
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k A
= = = =A = (4)2 + (-2)2 + (3)2
= = = =A = 29 = 5.39 v
UNIT VECTORSVector Addition/Subtraction
Example 1. Given the following vectors:
a) B - A + C b) C - B - A
A = (-2 i + 3 j + 4 k)mi
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B = (3 i + j - 3 k)mi
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C = (-5 i - 2 j + 2 k)mi
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A B = (Ax i + Ay j + Az k) (Bx i + By j + Bz k)i
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A B = (Ax Bx) i + (Ay By) j + (Az Bz) ki
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MULTIPLICATION OF VECTORSVector by a Scalar
- The product is another vector and takes the direction of the given vector.
A b = C
a = 25 kg
B = 4 m, N10oE
C = (-5 i -2 j + 2 k)Ni
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Example 2. Given the following, determine (a) a B and (b) a C.
MULTIPLICATION OF VECTORSScalar or Dot Product
- The dot product between two vectors is a scalar quantity.
Where: A & B – magnitudes of the given vectors - lesser angle between vectors connected tail-to-tail
A B = c
A B = AB Cos α θ β φφα θ β
α θ β φα θ β φ
φα θ βα θ β φ
Cases:i. 0o 90o - Parallel or Acute Angle; A B is (+).α θ β φ
φα θ βα θ β φ
<
≤
>
≥
<
≤
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≥
ii. 90o 180o - Anti-parallel or Obtuse Angle; A B is (-).α θ β φφα θ β
α θ β φ
<
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≥
<
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≥iii. = 90o - Perpendicular; A B is zero(0).α θ β φφα θ β
α θ β φ
MULTIPLICATION OF VECTORSScalar or Dot Product
Example 3. Given the following, determine the scalar product between the given vector quantities.
A = 25 v, 25o S of W B = 40 v, N10oE
MULTIPLICATION OF VECTORSScalar or Dot Product
0 0+1
A B = (Ax i + Ay j + Az k) (Bx i + By j + Bz k)i
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Multiplying, first, vector A = Ax i + Ay j + Az k to Bx i , it will give us the following:
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k(Ax i + Ay j + Az k) Bx ii
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= AxBx(i i) + AyBx(j i) + AzBx(k i)
(i i) = 1(1) Cos ; = 0o ; Cos 0 = +1i
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α θ β φφα θ β
α θ β φ
α θ β φφα θ β
α θ β φ(i i) = +1 = (j j) = (k k)i
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i (-i) = 1(1) Cos ; = 180o ; Cos 180 = -1i
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α θ β φφα θ β
α θ β φ
α θ β φφα θ β
α θ β φi (-i) = -1 = j (-j) = k (-k)i
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i j = 1(1) Cos ; = 90o ; Cos 90 = 0i
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α θ β φφα θ β
α θ β φ
α θ β φφα θ β
α θ β φi j = 0 = j i = i k = k i = j k = k ji
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MULTIPLICATION OF VECTORSScalar or Dot Product
It follows that:
Therefore!
0 0+1
A B = (Ax i + Ay j + Az k) (Bx i + By j + Bz k)i
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Multiplying, first, vector A = Ax i + Ay j + Az k to Bx i , it will give us the following:
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k(Ax i + Ay j + Az k) Bx ii
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= AxBx(i i) + AyBx(j i) + AzBx(k i)
(Ax i + Ay j + Az k) Bx i = AxBxi
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k(Ax i + Ay j + Az k) By j = AyByi
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(Ax i + Ay j + Az k) Bz i = AzBzi
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kA B = Ax Bx + Ay By + Az Bz
MULTIPLICATION OF VECTORSScalar or Dot Product
Example 4. Determine a) dot product and b) angle between the given vectors:
A = (-2 i + 3 j + 4 k)mi
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B = (-5 i - 2 j + 2 k)Ni
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MULTIPLICATION OF VECTORSVector or Cross Product
- The cross product between two vectors is another vector quantity.
- Magnitude
Example 5. Given the following, determine the magnitude and direction vector product between the given vector quantities.
A x B = C
A x B = AB Sin α θ β φφα θ β
α θ β φWhere: A & B – magnitudes of the given vectors - lesser angle between vectors connected tail-to-tailα θ β φ
φα θ βα θ β φ
A = 25 v, 25o S of W B = 40 v, N10oE
A x B = C
MULTIPLICATION OF VECTORSVector or Cross Product
Right-Hand Rule: Direction
- The index finger points to the direction of the first vector.
- The middle finger points to the direction of the second vector.
- The thumb points to the direction of the product.
First Vector Second Vector
Product
- First & second vectors lie on one plane and the product is perpendicular to the plane.
A
B
C
= AxBx(i x i) + AyBx(j x i) + AzBx(k x i)i
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MULTIPLICATION OF VECTORSVector or Cross Product
0
A x B = (Ax i + Ay j + Az k) x (Bx i + By j + Bz k)i
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Multiplying, first, vector A = Ax i + Ay j + Az k to Bx i , it will give us the following:
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k(Ax i + Ay j + Az k) x Bx ii
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k(i x i) = 1(1) Sin ; = 0o ; Sin 0 = 0i
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α θ β φφα θ β
α θ β φ
α θ β φφα θ β
α θ β φ(i x i) = 0 = (j x j) = (k x k)i
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ki x (-i) = 1(1) Sin ; = 180o ; Sin 180 = 0i
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α θ β φφα θ β
α θ β φ
α θ β φφα θ β
α θ β φi x (-i) = 0 = j x (-j) = k x (-k)i
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MULTIPLICATION OF VECTORSVector or Cross Product
= AxBx(i x i) + AyBx(j x i) + AzBx(k x i)i
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0
A x B = (Ax i + Ay j + Az k) x (Bx i + By j + Bz k)i
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Multiplying, first, vector A = Ax i + Ay j + Az k to Bx i , it will give us the following:
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k(Ax i + Ay j + Az k) x Bx ii
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i x j = 1(1) Sin ; = 90o; Sin 90 = +1i
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α θ β φφα θ β
α θ β φ
α θ β φφα θ β
α θ β φi x j = +ki
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i x j = +1, along (+)z - axisi
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j x i = -ki
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j x i = +1, along (-)z - axisi
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j x k = +ii
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j x k = +1, along (+)x - axisi
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k x j = -ii
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k x j = +1, along (-)x - axisi
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k x i = +ji
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k x i = +1, along (+)y - axisi
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i x k = - ji
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i x k = +1, along (-)y - axisi
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-k j
= AxBx(i x i) + AyBx(j x i) + AzBx(k x i)i
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Multiplying, first, vector A = Ax i + Ay j + Az k to Bx i , it will give us the following:
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MULTIPLICATION OF VECTORSVector or Cross Product
0
A x B = (Ax i + Ay j + Az k) x (Bx i + By j + Bz k)i
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k(Ax i + Ay j + Az k) x Bx ii
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kii
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-k j
A x B = (Ay Bz - Az By) i + (Az Bx - Ax Bz)j + (Ax By - Ay Bx)ki
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MULTIPLICATION OF VECTORSVector or Cross Product
Using Matrix
A x B = (Ax i + Ay j + Az k) x (Bx i + By j + Bz k)i
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Ax
Bx
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Ay
By
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Az
Bz
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Ax
Bx
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A x B = + + (Ax By - Ay Bx)ki
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(Az Bx - Ax Bz)ji
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(Ay Bz - Az By) ii
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MULTIPLICATION OF VECTORSVector or Cross Product
Example 6. Determine a) cross product(magnitude & direction) and b) angle between the given vectors.
A = (-2 i + 3 j + 4 k)mi
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B = (-5 i - 2 j + 2 k)Ni
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