Test Index

Motion in a Plane

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Question : 22
Total: 32
^
i
 and
^
j
 are unit vectors along x and y-axis respectively. What is the magnitude and direction of the vectors
^
i
+
^
j
 and
^
i
−
^
j
? What are the components of a vector A=2
^
i
+3
^
j
 along the directions of
^
i
+
^
j
 and
^
i
−
^
j
 ?
Solution:  
(a) Magnitude of (
^
i
+
^
j
)=|
^
i
+
^
j
| =√(1)2+(1)2=√2
Let the vector (
^
i
+
^
j
 ) makes an angle θ with the direction of
^
i
 , then
cos‌theta=
(
^
i
+
^
j
).
^
i
|
^
i
+
^
j
||
^
i
|
 =
1
(√2)(1)
=
1
√2
=cos‌45°
or θ = 45°
Magnitude of (
^
i
−
^
j
)=|
^
i
−
^
j
| =√(1)2+(−1)2=√2
Similarly, if θ is the angle which the vector (
^
i
−
^
j
 ) makes with the direction of
^
i
 , then
cos‌theta=
(
^
i
−
^
j
).
^
i
|
^
i
−
^
j
||
^
i
|
 =
1
√2
=cos‌45°
or θ = 45°
Here, θ=−45° with
^
i
.
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