the magnitude of vector product of unit victors is perpendicular to each other is, ...............
Answers
Answer:
The vector product of two vectors is a vector perpendicular to both of them. Its magnitude is obtained by multiplying their magnitudes by the sine of the angle between them. The direction of the vector product can be determined by the corkscrew right-hand rule.
Physics > System of Particles and Rotational Dynamics > Vector Product of Two Vectors
System of Particles and Rotational Dynamics
Vector Product of Two Vectors
We know that a vector has magnitude as well as a direction. But do we know how any two vectors multiply? Let us now study about the cross product of these vectors in detail.
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Solve
Questions
Calculate the cross product between a=(3, -3, 1)a=(3,−3,1) and b=(4, 9, 2)b=(4,9,2).
1 Verified answer
If |\vec{a}\times \vec{b}|∣
a
×
b
∣ =16, |\vec{a}|∣
a
∣ = 10 and \vec{a} . \vec{b}
a
.
b
=12 find |\vec{b}|∣
b
∣.
1 Verified answer
If \overline{a}, \overline{b}, \overline{c}
a
,
b
,
c
and \overline{a'}, \overline{b'}, \overline{c'}
a
′
,
b
′
,
c
′
are reciprocal system of vectors, then both statement is ?
(a) \overline{a'} \times \overline{b'} + \overline{b'} \times \overline{c'} + \overline{c'} \times \overline{a'}=\dfrac{\overline{a}+\overline{b}+\overline{c}}{[\overline{a} \overline{b} \overline{c}]}
a
′
×
b
′
+
b
′
×
c
′
+
c
′
×
a
′
=
[
a
b
c
]
a
+
b
+
c
(b) (\overline{a}+\overline{b}+\overline{c})\cdot (\overline{a'}+\overline{b'}+\overline{c'})=3(
a
+
b
+
c
)⋅(
a
′
+
b
′
+
c
′
)=3
1 Verified answer
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Vector Product of Two Vectors
Vector product also means that it is the cross product of two vectors.
Cross Product
If you have two vectors a and b then the vector product of a and b is c.
c = a × b
So this a × b actually means that the magnitude of c = ab sinθ where θ is the angle between a and b and the direction of c is perpendicular to a well as b. Now, what should be the direction of this cross product? So to find out the direction, we use the rule which we call it as the ”right-hand thumb rule”.
Suppose we want to find out the direction of a × b here we curl our fingers from the direction of a to b. So if we curl our fingers in a direction as shown in the above figure, your thumb points in the direction of c that is in an upward direction. This thumb denotes the direction of the cross product.
While applying rules to direction, the rotation should be taken to smaller angles that is <180° between a and b. So the fingers should always be curled in acute angle between a and b.
Properties of Vector Cross Product
1] Vector product is not commutative. That means a × b ≠ b × a
We saw that a × b = c here the thumb is pointing in an upward direction. Whereas in b × a the thumb will point in the downward direction. So, b × a = – c. So it is not commutative.
2] There is no change in the reflection.
What happens to a × b in the reflection? Suppose vector a goes and strikes the mirror, so the direction of a will become – a. So under reflection, a will become – a and b will become – b. Now a × b will become -a × -b = a × b
3] It is distributive with respect to vector addition.
This means that if a × ( b + c ) = a × b + a × c. This is true in case of addition.
Vector Product of Unit Vectors
The three unit vectors are \hat{i}
i
^
, \hat{j}
j
^
and \hat{k}
k
^
. So,
\hat{i}
i
^
× \hat{i}
i
^
= 0
\hat{i}
i
^
× \hat{j}
j
^
= 1 \hat{k}
k
^
\hat{i}
i
^
× \hat{k}
k
^
=1 – \hat{j}
j
^
\hat{j}
j
^
× \hat{i}
i
^
= – \hat{k}
k
^
\hat{j}
j
^
× \hat{j}
j
^
= 0
\hat{j}
j
^
× \hat{k}
k
^
= 1 \hat{i}
i
^
\hat{k}
k
^
× \hat{i}
i
^
= \hat{j}
j
^
\hat{k}
k
^
× \hat{j}
j
^
= - \hat{i}
i
^
\hat{k}
k
^
× \hat{k}
k
^
= 0
This is how we determine the vector product of unit vectrors.
Mathematical Form of Vector Product
a = ax \hat{i}
i
^
+ ay \hat{j}
j
^
+az \hat{k}
k
^
b = bx \hat{i}
i
^
+ by \hat{j}
j
^
+bz \hat{k}
k
^
a×b = ( ax \hat{i}
i
^
+ ay \hat{j}
j
^
+az \hat{k}
k
^
) × ( bx \hat{i}
i
^
+ by \hat{j}
j
^
+bz \hat{k}
k
^
)
= ax \hat{i}
i
^
× ( bx \hat{i}
i
^
+ by \hat{j}
j
^
+bz \hat{k}
k
^
) + ay \hat{j}
j
^
× ( bx \hat{i}
i
^
+ by \hat{j}
j
^
+bz \hat{k}
k
^
) + az \hat{k}
k
^
× ( bx \hat{i}
i
^
+ by \hat{j}
j
^
+bz \hat{k}
k
^
)
= ax by \hat{k}
k
^
– ax bz \hat{j}
j
^
+ ay bz \hat{i}
i
^
+ az bx \hat{j}
j
^
– az by \hat{i}
i
^
a×b = (ay bz – az by) \hat{i}
i
^
+ (az bx – ax bz) \hat{j}
j
^
+ (ax by – ay bx) \hat{k}
k
^
So the determinant form of the vectors will be, a×b =
\hat{i}
i
^
\hat{j}
j
^
\hat{k}
k
^
ax ay az
bx by bz
Solved Question
Q1. The magnitude of the vector product of two vectors \vec{P}
P
and \vec{Q}
Q
may be:
Equal to PQ
Less than PQ
Equal to zero
All of the above.
Answer: The correct option is “D”. | \vec{P}
P
× \vec{Q}
Q
= \vec{P}
P
\vec{Q}
Q
sinθ, where θ is the angle between P and Q.
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