Find unit vector along the vector
À = 3i^+4j^ +k^. show that vectors
3i^- 4j^+k^ and B`= 9i^-12j^+3k^ have
same directions.
Answers
Given :-
◉ A = 3i + 4j + k
◉ B = 9i - 12j + 3k
◉ C = 3i - 4j + k
To Find or Show :-
◉ A unit vector along the vector A
◉ Vectors A and B have same directions
Solution :-
For finding unit vector along a given vector, we use the general formula:
⇒ u = A / |A|
Where,
- u = unit vector
- A = any given vector
- |A| = Magnitude of vector A
So, We have to find the magnitude of A first
⇒ |A| = √(3² + 4² + 1²)
⇒ |A| = √(9 + 16 + 1)
⇒ |A| = √26
Now, Substituting the values
⇒ u = (3i + 4j + k) / √26
⇒ u = 3/√26 i + 4/√26 j + 1/√26 k
⇒ u = 3√26/26 i + 4√26/26 j + √26/26 k
Hence, u is the required unit vector.
Now, Let us find whether vectors C and B have same directions,
We have
- C = 3i - 4j + k
- B = 9i - 12j + 3k
If we closely observe vector B is same as vector A in direction but is 3 times in magnitude. How did I find it out? Let's see
⇒ B = 9i - 12j + 3k
⇒ B = 3(3i - 4j + k)
⇒ B = 3A
Hence, we conclude that vector C and B have same directions.
Answer:
We have:
3
i
+
4
j
−
k
Let
u
=
3
i
+
4
j
−
k
.
Unit vectors are of the form
ˆ
u
=
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|
u
|
:
⇒
ˆ
u
=
3
i
+
4
j
−
k
|
3
i
+
4
j
−
k
|
⇒
ˆ
u
=
3
i
+
4
j
−
k
√
3
2
+
4
2
+
(
−
1
)
2
⇒
ˆ
u
=
3
i
+
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j
−
k
√
9
+
16
+
1
⇒
ˆ
u
=
3
i
+
4
j
−
k
√
26