How to resolve a vector into its parallel and perpendicular components?Give one example also.
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Components of a Vector Which are Parallel and Perpendicular to a Given Vector
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COMPONENTS OF VECTOR PARALLEL/PERPENDICULAR TO ANOTHER VECTOR - DEFINITION
The components of b along and perpendicular to a are (
∣a∣
2
a.b
)a and b−(
∣a∣
2
a.b
)a respectively
CONDITION FOR PARALLEL AND PERPENDICULAR LINES - RESULT
Let
a
=a
1
i
^
+a
2
j
^
+a
3
k
^
and
b
=b
1
i
^
+b
2
j
^
+b
3
k
^
1. If
a
and
b
are perpendicular then their dot product is zero
i.e.
a
⋅
b
=0
⇒(a
1
i
^
+a
2
j
^
+a
3
k
^
)⋅(b
1
i
^
+b
2
j
^
+b
3
k
^
)=0
⇒a
1
b
1
+a
2
b
2
+a
3
b
3
=0
2. If
a
and
b
are parallel
Then, a
1
=kb
1
, a
2
=kb
2
and a
3
=kb
3
⇒
b
1
a
1
=
b
2
a
2
=
b
3
a
3
COMPONENTS OF PARALLEL AND PERPENDICULAR VECTORS - FORMULA
Let
a
=a
1
i
^
+a
2
j
^
+a
3
k
^
and
b
=b
1
i
^
+b
2
j
^
+b
3
k
^
1. If
a
and
b
are perpendicular then their dot product is zero
i.e.
a
⋅
b
=0
⇒(a
1
i
^
+a
2
j
^
+a
3
k
^
)⋅(b
1
i
^
+b
2
j
^
+b
3
k
^
)=0
⇒a
1
b
1
+a
2
b
2
+a
3
b
3
=0
2. If
a
and
b
are parallel
Then, a
1
=kb
1
, a
2
=kb
2
and a
3
=kb
3
⇒
b
1
a
1
=
b
2
a
2
=
b
3
a
3
COMPONENTS OF A VECTOR PARALLEL/PERPENDICULAR TO ANOTHER VECTOR - EXAMPLE
Example:- If AB⊥BC, then the value of k is if A(k,1,−1), B(2k,0,2), C(2+2k,k,1)
Solution:- Given, A(k,1,−1),B(2k,0,2),C(2+2k,k,1)
AB
=(k,−1,3)
BC
=(2,k,−1)
AB is perpendicular to BC
So,
AB
.
BC
=0
2k−k−3=0
∴k=3