PHY10-Unit Vector.pdf

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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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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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


<

≤

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≥

<

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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


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


0 0+1


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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 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 φ


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 φ


α θ β φi j = 0 = j i = i k = k i = j k = k ji

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It follows that:

Therefore!

0 0+1


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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


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


A = 25 v, 25o S of W B = 40 v, N10oE

A x B = C


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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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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k(i x i) = 1(1) Sin ; = 0o ; Sin 0 = 0i

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α θ β φ


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 φ


α θ β φi x (-i) = 0 = j x (-j) = k x (-k)i

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i x j = 1(1) Sin ; = 90o; Sin 90 = +1i

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α θ β φ


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


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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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Using Matrix


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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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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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Documents

PHY10-Unit Vector.pdf