Advertisements
Advertisements
Question
If \[\vec{c}\] s perpendicular to both \[\vec{a} \text{ and } \vec{b}\] then prove that it is perpendicular to both \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b}\]
Advertisements
Solution
\[\text{ Given that } \vec{c} \text{ is perpendicular to both } \vec{a} \text{ and } \vec{b} . \]
\[ \Rightarrow \vec{c} . \vec{a} = 0 \text{ and } \vec{c} . \vec{b} = 0 . . . \left( 1 \right)\]
\[\text{ Now },\]
\[ \vec{c} . \left( \vec{a} + \vec{b} \right) = \vec{c} . \vec{a} + \vec{c} . \vec{b} = 0 + 0 = 0................. \left[ \text{ From }\left( 1 \right) \right]\]
\[\text{ So }, \vec{c} \text{ is perpendicular to } \vec{a} + \vec{b} . \]
\[\text{ Again },\]
\[ \vec{c} . \left( \vec{a} - \vec{b} \right) = \vec{c} . \vec{a} - \vec{c} . \vec{b} = 0 - 0 = 0 ..................\left[ \text{ From }\left( 1 \right) \right]\]
\[\text{ So }, \vec{c} \text{ is perpendicular to } \vec{a} - \vec{b} . \]
APPEARS IN
RELATED QUESTIONS
Compute the magnitude of the following vector:
`veca = hati + hatj + hatk;` `vecb = 2hati - 7hatj - 3hatk`; `vecc = 1/sqrt3 hati + 1/sqrt3 hatj - 1/sqrt3 hatk`
Write two different vectors having same magnitude.
Write two different vectors having same direction.
The value of is `hati.(hatj xx hatk)+hatj.(hatixxhatk)+hatk.(hatixxhatj)` is ______.
If θ is the angle between any two vectors `veca` and `vecb,` then `|veca.vecb| = |veca xx vecb|` when θ is equal to ______.
If \[\vec{a} = 5 \hat{i} - \hat{j} - 3 \hat{k} \text{ and } \vec{b} = \hat{i} + 3 \hat{j} - 5 \hat{k} ,\] then show that the vectors \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} \] are orthogonal.
If two vectors \[\vec{a} \text{ and } \vec{b}\] are such that \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 1 \text{ and } \vec{a} \cdot \vec{b} = 1,\] then find the value of \[\left( 3 \vec{a} - 5 \vec{b} \right) \cdot \left( 2 \vec{a} + 7 \vec{b} \right) .\]
If \[\vec{a}\] is a unit vector, then find \[\left| \vec{x} \right|\] in each of the following.
\[\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 12\]
Find \[\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|\] if
\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 12 \text{ and } \left| \vec{a} \right| = 2\left| \vec{b} \right|\]
Find \[\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|\] if
\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 8 \text{ and } \left| \vec{a} \right| = 8\left| \vec{b} \right|\]
Find \[\left| \vec{a} \right| and \left| \vec{b} \right|\] if
\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 3\text{ and } \left| \vec{a} \right| = 2\left| \vec{b} \right|\]
Find \[\left| \vec{a} - \vec{b} \right|\] if
\[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 4\]
Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] if
\[\left| \vec{a} \right| = \sqrt{3}, \left| \vec{b} \right| = 2 \text{ and } \vec{a} \cdot \vec{b} = \sqrt{6}\]
Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\]
\[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 1\]
Express the vector \[\vec{a} = 5 \text{i} - 2 \text{j} + 5 \text{k}\] as the sum of two vectors such that one is parallel to the vector \[\vec{b} = 3 \text{i} + \text{k}\] and other is perpendicular to \[\vec{b}\]
If \[\vec{a} \text{ and } \vec{b}\] are two vectors of the same magnitude inclined at an angle of 30°, such that \[\vec{a} \cdot \vec{b} = 3, \text{ find } \left| \vec{a} \right|, \left| \vec{b} \right| .\]
Express \[2 \hat{i} - \hat{j} + 3 \hat{k}\] as the sum of a vector parallel and a vector perpendicular to \[2 \hat{i} + 4 \hat{j} - 2 \hat{k} .\]
Decompose the vector \[6 \hat{i} - 3 \hat{j} - 6 \hat{k}\] into vectors which are parallel and perpendicular to the vector \[\hat{i} + \hat{j} + \hat{k} .\]
Let \[\vec{a} = 5 \hat{i} - \hat{j} + 7 \hat{k} \text{ and } \vec{b} = \hat{i} - \hat{j} + \lambda \hat{k} .\] Find λ such that \[\vec{a} + \vec{b}\] is orthogonal to \[\vec{a} - \vec{b}\]
If \[\vec{a} \cdot \vec{a} = 0 \text{ and } \vec{a} \cdot \vec{b} = 0,\] what can you conclude about the vector \[\vec{b}\] ?
If \[\vec{a,} \vec{b,} \vec{c}\] are three non-coplanar vectors, such that \[\vec{d} \cdot \vec{a} = \vec{d} \cdot \vec{b} = \vec{d} \cdot \vec{c} = 0,\] then show that \[\vec{d}\] is the null vector.
If a vector \[\vec{a}\] is perpendicular to two non-collinear vectors \[\vec{b} \text{ and } \vec{c} , \text{ then show that } \vec{a}\] is perpendicular to every vector in the plane of \[\vec{b} \text{ and } \vec{c} .\]
If \[\vec{a} + \vec{b} + \vec{c} = \vec{0} ,\] show that the angle θ between the vectors \[\vec{b} \text{ and } \vec{c}\] is given by \[\frac{\left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 - \left| \vec{c} \right|^2}{2\left| \vec{b} \right| \left| \vec{c} \right|} .\]
Let \[\vec{u,} \vec{v} \text{ and } \vec{w}\] be vectors such \[\vec{u} + \vec{v} + \vec{w} = \vec{0} .\] If \[\left| \vec{u} \right| = 3, \left| \vec{v} \right| = 4 \text{ and } \left| \vec{w} \right| = 5,\] then find \[\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} .\]
Find the values of x and y if the vectors \[\vec{a} = 3 \hat{i} + x \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + \hat{j} + y \hat{k}\] are mutually perpendicular vectors of equal magnitude.
What is the product of `(3veca * 5vecb) * (2veca + 7vecb)`
When a vector is multiplied by a scalar (a numerical value), what is the nature of the resulting quantity?
In which of the following fields are vector operations NOT commonly used?
