Advertisements
Advertisements
Question
Can you add three unit vectors to get a unit vector? Does your answer change if two unit vectors are along the coordinate axes?
Advertisements
Solution
Yes we can add three unit vectors to get a unit vector.
No, the answer does not change if two unit vectors are along the coordinate axes. Assume three unit vectors \[\hat{i,} - \hat { i} \text { and }\hat { j}\] along the positive x-axis, negative x-axis and positive y-axis, respectively. Consider the figure given below:
The magnitudes of the three unit vectors ( \[\hat {i,} - \hat { i }\text { and } \hat {j})\] are the same, but their directions are different.
So, the resultant of \[\hat { i} \text { and } - \hat {i}\] is a zero vector.
Now, \[\hat {j} + \vec{0} = \hat {j}\] (Using the property of zero vector)
∴ The resultant of three unit vectors ( \[\hat{i,} - \hat {i} \text { and }\hat { j }\]) is a unit vector (\[\hat {j}\]).
APPEARS IN
RELATED QUESTIONS
Some of the most profound statements on the nature of science have come from Albert Einstein, one of the greatest scientists of all time. What do you think did Einstein mean when he said : “The most incomprehensible thing about the world is that it is comprehensible”?
“Every great physical theory starts as a heresy and ends as a dogma”. Give some examples from the history of science of the validity of this incisive remark
A unitless quantity
The dimensions ML−1 T−2 may correspond to
Choose the correct statements(s):
Find the dimensions of linear momentum .
Find the dimensions of magnetic permeability \[\mu_0\]
The relevant equation are \[F = qE, F = qvB, \text{ and }B = \frac{\mu_0 I}{2 \pi a};\]
where F is force, q is charge, v is speed, I is current, and a is distance.
Find the dimensions of the coefficient of linear expansion α and
Is a vector necessarily changed if it is rotated through an angle?
Can you add two vectors representing physical quantities having different dimensions? Can you multiply two vectors representing physical quantities having different dimensions?
Let ε1 and ε2 be the angles made by \[\vec{A}\] and -\[\vec{A}\] with the positive X-axis. Show that tan ε1 = tan ε2. Thus, giving tan ε does not uniquely determine the direction of \[\vec{A}\].
Can you have \[\vec{A} \times \vec{B} = \vec{A} \cdot \vec{B}\] with A ≠ 0 and B ≠ 0 ? What if one of the two vectors is zero?
Add vectors \[\vec{A} , \vec{B} \text { and } \vec{C}\] each having magnitude of 100 unit and inclined to the X-axis at angles 45°, 135° and 315° respectively.
Refer to figure (2 − E1). Find (a) the magnitude, (b) x and y component and (c) the angle with the X-axis of the resultant of \[\overrightarrow{OA}, \overrightarrow{BC} \text { and } \overrightarrow{DE}\].
Suppose \[\vec{a}\] is a vector of magnitude 4.5 units due north. What is the vector (a) \[3 \vec{a}\], (b) \[- 4 \vec{a}\] ?
Let A1 A2 A3 A4 A5 A6 A1 be a regular hexagon. Write the x-components of the vectors represented by the six sides taken in order. Use the fact the resultant of these six vectors is zero, to prove that
cos 0 + cos π/3 + cos 2π/3 + cos 3π/3 + cos 4π/3 + cos 5π/3 = 0.
Use the known cosine values to verify the result.
Prove that \[\vec{A} . \left( \vec{A} \times \vec{B} \right) = 0\].
Round the following numbers to 2 significant digits.
(a) 3472, (b) 84.16. (c)2.55 and (d) 28.5
If π = 3.14, then the value of π2 is ______
