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Dot and Cross Product
VECTORS
DOT PRODUCT OR SCALAR PRODUCT
Multiplication of vectors
A vector multiplied by another vector may give a scalar or a vector. Hence there are two types of products for multiplication of two vectors.
a) dot product or scalar product
b) cross product or vector product
Scalar Product or Dot Product
The scalar product of two vectors \(\vec P\) and \(\vec Q\) is defined as the product of the magnitudes of \(\vec P\) and \(\vec Q\) and of the cosine of the angle q formed by \(\vec P\) and \(\vec Q\).
The scalar product of \(\vec P\) and \(\vec Q\) is denoted by \(\vec P\).\(\vec Q\)
such that \( \overrightarrow P \,.\,\overrightarrow Q = \left| {\overrightarrow P } \right|\left| {\overrightarrow Q } \right|\cos \theta = PQ\cos \theta \)
\( P\cos \theta \) is component of \( \overrightarrow P \,\,\text{along}\,\,\overrightarrow Q \) and \( Q\cos \theta \) is component of \( \overrightarrow Q \,\,\text{along}\,\,\overrightarrow P \)
The dot product of two vectors is a scalar.
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Dot and Cross Product
Laws of Scalar Product:
a) Scalar product obeys commutative law i.e. \( \overrightarrow A .\overrightarrow B = \,\overrightarrow {B.} \overrightarrow A \)
b) Scalar product obeys distributive law i.e. \(\overrightarrow A(\overrightarrow B+\overrightarrow C)=\overrightarrow A.\overrightarrow B+\overrightarrow A.\overrightarrow C\)
c) It does not obey associative law. -
Dot and Cross Product
Properties of Scalar Product:
1) Scalar product of two parallel vectors is maximum
\( \overrightarrow A .\overrightarrow B = \left| {\overrightarrow A } \right|\left| {\overrightarrow B } \right|cos\theta = AB (\because \theta = 0^0 ) \)
2) The scalar product of two opposite vectors is negative and minimum
3) The scalar product of two perpendicular vectors is zero (or) If the scalar product of two nonzero vectors vanishes then the vectors are orthogonal.
i.e. \(\overrightarrow A.\overrightarrow B=ABcos\theta=0(\therefore \theta =90^0)\)
4) Scalar product is negative if 90° < q < 180°
5) In case of orthogonal unit vectors \( \hat i.\hat i = \hat j.\hat j = \hat k.\hat k = 1 \) ; \( \hat i.\hat j = \hat j.\hat k = \hat k.\hat i = 0 \)
6) In terms of Components of vectors, If \( \overrightarrow A = A_x \hat i + A_y \hat j + A_z \hat k\, \) and \( \overrightarrow B = B_x \hat i + B_y \hat j + B_z \hat k\, \) Then, \(\overrightarrow A.\overrightarrow B=A_xB_x+A_yB_y+A_zB_z \)
for perpendicular vectors \( \text{A}_\text{x} \text{B}_\text{x} \text{ + A}_\text{y} \text{B}_\text{y} \text{ + A}_\text{z} \text{B}_\text{z} = 0 \)
7) If \( \overrightarrow A = A_x \hat i + A_y \hat j + A_z \hat k\, \), then \( \overrightarrow A \overrightarrow {.A } = A^2 _\text{x} \text{ + A}^2 _\text{y} \text{ + A}^2 _\text{z} = \,A^2 \)
Magnitude of any vector is \( \left| {\vec A} \right|\text{ = A}\,\text{ = }\,\sqrt {\text{A}^2 _\text{x} \text{ + A}^2 _\text{y} \text{ + A}^2 _\text{z} } \) -
Dot and Cross Product
Applications of Dot Product
i) Angle between two vectors \(\overrightarrow A\) and \(\overrightarrow B\) can be found from \( cos \theta = \frac {\overrightarrow A .\overrightarrow B } {| \overrightarrow A || {\overrightarrow B } |} \)
ii) Component of \(\overrightarrow A\) along \(\overrightarrow B\) = A cos\(\theta\) = \( \frac{{\vec A.\vec B}} {B} \) =\(\overrightarrow A \).\(\hat B\) and
Vector component of \(\overrightarrow A\) along \(\overrightarrow B\) is \( \frac{{\vec A\,.\,\vec B}} {B}\,\hat B = \left( {\vec A\,.\,\hat B} \right)\,\hat B \)
iii) Component of \(\overrightarrow B\) along \(\overrightarrow A\) = B cos \(\theta\) =\( \frac{{\vec A.\vec B}} {A} \) =\(\hat A\).\(\overrightarrow B\) and
Vector component of \(\overrightarrow A\) along \(\overrightarrow B\) is \( \frac{{\vec A\,.\,\vec B}} {A}\hat A = \left( {\vec B\,.\,\hat A} \right)\,\hat A \)
iv) The component of \(\overrightarrow A\) perpendicular to \(\overrightarrow B\) in the same plane is \( \overrightarrow C = \overrightarrow A - \left( {\overrightarrow A \,.\,\hat B} \right)\hat B \)
v) The component of \(\overrightarrow A\) perpendicular to \(\overrightarrow B\) in the same plane is \( \overrightarrow D = \overrightarrow B - \left( {\overrightarrow B \,.\,\hat A} \right)\hat A \)
vi) \( \left| {\vec A + \vec B} \right| = \sqrt {\vec A.\vec A + \vec B.\vec B + \vec A.\vec B + \vec B.\vec A} \) \( = \sqrt {A^2 + B^2 + 2AB\cos \theta } \) -
Dot and Cross Product
Examples of dot product :
1) Work done is the dot product of force and displacement
W = \( \vec F.\,\vec S = FS\cos \theta \).
2) Power is the dot product of force and instantaneous velocity
Power P = \( \vec F.\,\vec V = FV\cos \theta \)
3) Magnetic Flux \( \phi = \,\overrightarrow B \,.\,\overrightarrow A \) etc.