2. When is it true that u+ v=v+ u
Answers
Answer:
ANSWER
Given two vectors,
u
and
v
.
Let the angle between them be θ.
To Prove (i) (
u
.
v
)
2
+(
u
×
v
)
2
=∣
u
∣
2
∣
v
∣
2
Dot product can be calculated as
u
.
v
=∣
u
∣∣
v
∣cosθ
On squaring the equation we get
⇒(
u
.
v
)
2
=∣
u
∣
2
∣
v
∣
2
cos
2
θ →(1)
Cross product can be calculated as
u
×
v
=∣
u
∣∣
v
∣sinθ
n
^
,where
n
^
is a unit vector.
On squaring the equation we get
⇒ (
u
×
v
)
2
=∣
u
∣
2
∣
v
∣
2
sin
2
θ →(2)
Adding equations 1 and 2 we get,
(
u
.
v
)
2
+(
u
×
v
)
2
= ∣
u
∣
2
∣
v
∣
2
(sin
2
θ+cos
2
θ)
⇒ (
u
.
v
)
2
+(
u
×
v
)
2
=∣
u
∣
2
∣
v
∣
2
Hence proved.
To Prove (ii) (
u
+1)
2
+(
v
+1)
2
=(1−
u
.
v
)
2
+∣
u
+
v
+(
u
×
v
)∣
2
Consider the Right Hand Side of the equation,
(1−
u
.
v
)
2
+∣
u
+
v
+(
u
×
v
)∣
2
=1+(
u
.
v
)
2
−2(
u
.
v
)+
u
2
+
v
2
+(
u
×
v
)
2
+2(
u
.
v
)+2
u
.(
u
×
v
)+2
v
.(
u
×
v
)
Note:
u
.(
u
×
v
) and
v
.(
u
×
v
) will be equal to zero as
u
×
v
is perpendicular to both
u
and
v
.
=1+(
u
.
v
)
2
+
u
2
+
v
2
+(
u
×
v
)
2
On rearranging the terms we get
=1+
u
2
+
v
2
+(
u
.
v
)
2
+(
u
×
v
)
2
=1+
u
2
+
v
2
+∣
u
∣
2
∣
v
∣
2
=(1+
u
2
)(1+
v
2
)
Hence proved.
Answered By
toppr
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