Advertisements
Advertisements
Question
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 \[\vec{a} - \vec{b}\]
Solution
\[\text{ Given that }\]
\[ \vec{a} = 5 \hat{i} - \text{j} + 7 \hat{k} ; \vec{b} = \hat{i} - \hat{j} + \lambda \hat{k} \]
\[ \therefore \vec{a} + \vec{b} = 5 \hat{i} - \hat{j} + 7 \hat{k} + \hat{i} - \hat{j} + \lambda \hat{k} = 6 \hat{i} - 2 \hat{j} + \left( 7 + \lambda \right) \hat{k} \]
\[\text{ and } \vec{a} - \vec{b} = 5 \hat{i} - \hat{j} + 7 \hat{k} - \left( \hat{i} - \hat{j} + \lambda \hat{k} \right) = 4 \hat{i} + 0 \hat{j} + \left( 7 - \lambda \right) \hat{k} \]
\[\text{ Given that } \vec{a} + \vec{b} \text{ is orthogonal to } \vec{a} - \vec{b} . \]
\[ \Rightarrow \left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 0\]
\[ \Rightarrow \left[ 6\hat{i} - 2 \hat{j} + \left( 7 + \lambda \right) \hat{k} \right] . \left[ 4 \hat{i} + 0 \hat{j} + \left( 7 - \lambda \right) \hat{k} \right] = 0\]
\[ \Rightarrow 24 + 0 + 49 - \lambda^2 = 0\]
\[ \Rightarrow \lambda^2 = 73\]
\[ \Rightarrow \lambda = \sqrt{73}\]
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 direction.
The value of is `hati.(hatj xx hatk)+hatj.(hatixxhatk)+hatk.(hatixxhatj)` is ______.
Find the projection of \[\vec{b} + \vec{c} \text { on }\vec{a}\] where \[\vec{a} = 2 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - 2 \hat{k} \text{ and } \vec{c} = 2 \hat{i} - \hat{j} + 4 \hat{k} .\]
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.
A unit vector \[\vec{a}\] makes angles \[\frac{\pi}{4}\text{ and }\frac{\pi}{3}\] with \[\hat{i}\] and \[\hat{j}\] respectively and an acute angle θ with \[\hat{k}\] . Find the angle θ and components of \[\vec{a}\] .
If \[\vec{a}\] is a unit vector, then find \[\left| \vec{x} \right|\] in each of the following.
\[\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 8\]
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 \[\left| \vec{a} - \vec{b} \right|\] if
\[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} \cdot \vec{b} = 8\]
Find \[\left| \vec{a} - \vec{b} \right|\]
\[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = 4 \text{ and } \vec{a} \cdot \vec{b} = 1\]
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 \[\hat{i} + \hat{j} + \hat{k} .\]
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 \[\left| \vec{a} \right| = a \text{ and } \left| \vec{b} \right| = b,\] prove that \[\left( \frac{\vec{a}}{a^2} - \frac{\vec{b}}{b^2} \right)^2 = \left( \frac{\vec{a} - \vec{b}}{ab} \right)^2 .\]
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 \[\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} .\]
Let \[\vec{a} = x^2 \hat{i} + 2 \hat{j} - 2 \hat{k} , \vec{b} = \hat{i} - \hat{j} + \hat{k} \text{ and } \vec{c} = x^2 \hat{i} + 5 \hat{j} - 4 \hat{k}\] be three vectors. Find the values of x for which the angle between \[\vec{a} \text{ and } \vec{b}\ \] is acute and the angle between \[\vec{b} \text{ and } \vec{c}\] is obtuse.
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.
If \[\vec{a}\] \[\vec{b}\] are two vectors such that \[\left| \vec{a} + \vec{b} \right| = \left| \vec{b} \right|\] then prove that \[\vec{a} + 2 \vec{b}\] is perpendicular to \[\vec{a}\]
What is the product of `(3veca * 5vecb) * (2veca + 7vecb)`
What are the values of x for which the angle between the vectors? `2x^2hati + 3xhatj + hatk` and `hati - 2hatj + x^2hatk` is obtuse?
