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VOOZH | about |
Solution:
We know that,
=>
=>
=>
=>
=>
Also,
=>
And
=>
=>
=>
=>
=>
=>
Solution:
Given
=>
=>, as is a unit vector.
=>
=>
=>
Solution:
Given that
=>
=>
=>
Using distributive property,
=>
If two vectors are parallel, then their cross-product is 0 vector.
=> and are parallel vectors.
=>
Hence proved.
Solution:
Given that,, and
We know that,
=>
=>
=>
=>
=>
=>
=>
=>
Solution:
Given, and
=>
=>
Either of the following conditions is true,
1.
2.
3.
4.is parallel to
=>
=>
Either of the following conditions is true,
1.
2.
3.
4. is perpendicular to
Since both these conditions are true, that implies atleast one of the following conditions is true,
1.
2.
3.
Solution:
Given, , and
As,
=>
=> is perpendicular to both and .
Similarly,
=> is perpendicular to both and
=> is perpendicular to both and
=> , and are mutually perpendicular.
As, , and are also unit vectors,
=> , and form an orthogonal right-handed triad of unit vectors
Hence proved.
Solution:
Given A(3, -1, 2), B(1, -1, 3) and C(4, -3, 1).
Let,
=>
=>
=>
Plane ABC has two vectors and
=>
=>
=>
=>
=>
=>
=>
=>
A vector perpendicular to both and is given by,
=>
=>
=>
=>
=>
To find the unit vector,
=>
=>
=>
=>
Solution:
Given that , and
From triangle law of vector addition, we have
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
Similarly,
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
Hence proved.
Solution:
Given, and
=>
=>
=>
=>
=>
Two vectors are perpendicular if their dot product is zero.
=>
=>
=>
=>
Hence proved.
Solution:
Given and forming an angle of .
Area of a parallelogram having diagonals and is
=>
=>
=>
Thus area is,
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area = square units
Solution:
We know that,
=>
=>
=>
=>
=>
=>
=>
=>
=>
Hence proved.
Solution:
Definition of : Let and be 2 non-zero, non-parallel vectors. Then , is defined as a vector with the magnitude of , and which is perpendicular to both the vectors and .
We know that,
=>
=>
=> .................(eq.1)
And as,
=>
=>
Substituting in (eq.1),
=>
=>
This set focuses on applying cross product properties and calculations in various scenarios.
Key points include:
1. Cross product properties (anticommutative, distributive, etc.)
2. Geometric applications (area, volume, perpendicularity)
3. Vector triple product and its expansion
4. Using cross products to solve geometric problems
1. If a = 3i - 2j + k, b = i + 2j - 3k, and c = 2i + j - k, find a · (b × c).
2. Prove that a × (b × c) + b × (c × a) + c × (a × b) = 0.
3. If a = i + 2j - k, b = 2i - j + 3k, and c = i + j + k, find the volume of the parallelepiped formed by these vectors.
4. Show that (a × b) × (c × d) = [a b d]c - [a b c]d.
5. Find a vector perpendicular to both a = 2i - j + 3k and b = i + 2j - k, and has a magnitude of 5 units.
6. Prove that (a × b) · (c × d) = (a · c)(b · d) - (a · d)(b · c).
7. If a, b, and c are unit vectors such that a + b + c = 0, prove that a × b = b × c = c × a.
8. Find the area of the triangle formed by the points A(1, -1, 2), B(3, 2, -1), and C(4, -3, 5).
9. Prove that |a × (b × c)| = |b||c|sin θ, where θ is the angle between b and c.
10. If a · b = 5, b · c = -2, c · a = 3, |a| = 2, |b| = 3, and |c| = 4, find |a × b + b × c + c × a|.
