Advertisements
Advertisements
Question
If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 0,\] find the relation between the magnitudes of \[\vec{a} \text{ and } \vec{b}\]
Advertisements
Solution
\[\text{ Given that }\]
\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 0\]
\[ \Rightarrow \left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 = 0\]
\[ \Rightarrow \left| \vec{a} \right|^2 = \left| \vec{b} \right|^2 \]
\[ \therefore \left| \vec{a} \right| = \left| \vec{b} \right|\]
APPEARS IN
RELATED QUESTIONS
If `veca and vecb` are two vectors such that `|veca+vecb|=|veca|,` then prove that vector `2veca+vecb` is perpendicular to vector `vecb`
Show that the vectors `veca, vecb` are coplanar if `veca+vecb, vecb+vecc ` are coplanar.
If `vec a=7hati+hatj-4hatk and vecb=2hati+6hatj+3hatk` , then find the projection of `vec a and vecb`
The scalar product of the vector `hati + hatj + hatk` with a unit vector along the sum of vectors `2hati + 4hatj - 5hatk` and `lambdahati + 2hatj + 3hatk` is equal to one. Find the value of `lambda`.
Find `lambda` if the scalar projection of `vec a = lambda hat i + hat j + 4 hat k` on `vec b = 2hati + 6hatj + 3hatk` is 4 units
Find \[\vec{a} \cdot \vec{b}\] when
\[\vec{a} = \hat{j} + 2 \hat{k} \text{ and } \vec{b} = 2 \hat{i} + \hat{k}\]
Find \[\vec{a} \cdot \vec{b}\] when
\[\vec{a} = \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + 3 \hat{j} - 2 \hat{k}\]
For what value of λ are the vectors \[\vec{a} \text{ and }\vec{b}\] perpendicular to each other if \[\vec{a} = \lambda \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{b} = 4\hat{i} - 9 \hat{j} + 2\hat{k}\]
For what value of λ are the vectors \[\vec{a} \text{ and } \vec{b}\] perpendicular to each other if
\[\vec{a} = \lambda \hat{i} + 2\hat{j} + \hat{k} \text{ and } \vec{b} = 5\hat{i} - 9 \hat{j} + 2\hat{k}\]
For what value of λ are the vectors \[\vec{a} \text{ and } \vec{b}\] perpendicular to each other if
\[\vec{a} = \lambda \hat{i} + 3 \hat{j} + 2 \hat{k}\text { and } \vec{b} = \hat{i} - \hat{j} + 3 \hat{k}\]
If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} \right| = 4, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 6\] find the angle between \[\vec{a} \text{ and } \vec{b} .\]
Find the cosine of the angle between the vectors \[4 \hat{i} - 3 \hat{j} + 3 \hat{k} \text{ and } 2 \hat{i} - \hat{j} - \hat{k} .\]
If the vectors \[3 \hat{i} + m \hat{j} + \hat{k} \text{ and } 2 \hat{i} - \hat{j} - 8 \hat{k}\] are orthogonal, find m.
If \[\vec{a} \text{ and } \vec{b}\] are vectors of equal magnitude, write the value of \[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) .\]
If \[\vec{b}\] is a unit vector such that\[\left( \vec{a} + \vec{b} \right) . \left( \vec{a} - \vec{b} \right) = 8, \text{ find } \left| \vec{a} \right| .\]
If \[\hat{a} , \hat{b}\] are unit vectors such that \[\hat{a} + \hat{b}\] is a unit vector, write the value of \[\left| \hat{a} - \hat{b} \right| .\]
If \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} . \vec{b} = 2, \text{ find } \left| \vec{a} - \vec{b} \right| .\]
If \[\vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + \hat{k} ,\] find the projection of \[\vec{a} \text{ on } \vec{b}\]
Write the component of \[\vec{b}\] along \[\vec{a}\]
Find the value of θ ∈(0, π/2) for which vectors \[\vec{a} = \left( \sin \theta \right) \hat{i} + \left( \cos \theta \right) \hat{j} \text{ and } \vec{b} = \hat{i} - \sqrt{3} \hat{j} + 2 \hat{k}\] are perpendicular.
Write the value of p for which \[\vec{a} = 3 \hat{i} + 2 \hat{j} + 9 \hat{k} \text{ and } \vec{b} = \hat{i} + p \hat{j} + 3 \hat{k}\] are parallel vectors .
Find λ when the projection of \[\vec{a} = \lambda \hat{i} + \hat{j} + 4 \hat{k} \text{ on } \vec{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k}\] is 4 units.
Write the projection of the vector \[7 \hat{i} + \hat{j} - 4 \hat{k}\] on the vector \[2 \hat{i} + 6 \hat{j}+ 3 \hat{k} .\]
Write the value of λ so that the vectors \[\vec{a} = 2 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}\] are perpendicular to each other.
If \[\vec{a}\] and \[\vec{b}\] are perpendicular vectors, \[\left| \vec{a} + \vec{b} \right| = 13\] and \[\left| \vec{a} \right| = 5\] find the value of \[\left| \vec{b} \right|\]
If \[\vec{a}\] and \[\vec{b}\] are two unit vectors such that \[\vec{a} + \vec{b}\] is also a unit vector, then find the angle between \[\vec{a}\] and \[\vec{b}\]
If \[\vec{a}\] and \[\vec{b}\] are unit vectors, then find the angle between \[\vec{a}\] and \[\vec{b}\] given that \[\left( \sqrt{3} \vec{a} - \vec{b} \right)\] is a unit vector.
If \[\vec{a} \text{ and } \vec{b}\] are two non-collinear unit vectors such that \[\left| \vec{a} + \vec{b} \right| = \sqrt{3},\] find \[\left( 2 \vec{a} - 5 \vec{b} \right) \cdot \left( 3 \vec{a} + \vec{b} \right) .\]
Let `veca, vecb, vecc` be three vectors of magnitudes 3, 4 and 5 respectively. If each one is petpendicular to the sum of the other two vectors, then `|veca + vecb + vecc|` =
If two vectors `veca` and `vecb` are such that `|veca|` = 2, `|vecb|` = 3 and `veca.vecb` = 4, then `|veca - 2vecb|` is equal to ______.
If `veca, vecb, vecc` are three non-zero unequal vectors such that `veca.vecb = veca.vecc`, then find the angle between `veca` and `vecb - vecc`.
Three vectors `veca, vecb` and `vecc` satisfy the condition `veca + vecb + vecc = vec0`. Evaluate the quantity μ = `veca.vecb + vecb.vecc + vecc.veca`, if `|veca|` = 3, `|vecb|` = 4 and `|vecc|` = 2.
If `veca.hati = veca.(hati + hatj) = veca.(hati + hatj + hatk)` = 1, then `veca` is ______.
If the two vectors `3hati + αhatj + hatk` and `2hati - hatj + 8hatk` are perpendicular to each other, then find the value of α.
If `veca = 2hati + hatj + 2hatk` and `vecb = 5hati - 3hatj + hatk`, find the projection of `vecb` on `veca`.
