87753
The projection of \(\vec{a}=2 \hat{i}+3 \hat{j}-2 \hat{k}\) on \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) is :
87754
A vector \(\vec{a}\) makes equal acute angles on the coordinate axis. Then the projection of vector \(\overrightarrow{\mathrm{b}}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) on \(\overrightarrow{\mathbf{a}}\) is
87755
If the vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{k}\) are the sides of a triangle \(\mathrm{ABC}\), then the length of the median through \(\mathrm{A}\) is
1 \(\sqrt{45}\)
2 \(\sqrt{18}\)
3 \(\sqrt{72}\)
4 \(\sqrt{33}\)
Explanation:
(D) : Given that, \(\overrightarrow{\mathrm{AB}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}\) \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) We know that, \(\overline{\mathrm{AM}}=\frac{1}{2}(\overline{\mathrm{AB}}+\overline{\mathrm{AC}})\) \(=\frac{1}{2}[(3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}})+(5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})]=\frac{8 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}}{2}\) \(=4 \hat{\mathrm{i}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) \(\therefore \mid \overline{\mathrm{AM}}=\sqrt{4^2+1^2+4^2}=\sqrt{33}\) Hence, the length of median is \(\sqrt{33}\).
COMEDK-2013
Vector Algebra
87756
If \(\vec{a}, \vec{b}, \vec{c}\) are three non-zero vectors such that each one of them are perpendicular to the sum of the other two vectors, then the value of \(|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}|^2\) is
87753
The projection of \(\vec{a}=2 \hat{i}+3 \hat{j}-2 \hat{k}\) on \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) is :
87754
A vector \(\vec{a}\) makes equal acute angles on the coordinate axis. Then the projection of vector \(\overrightarrow{\mathrm{b}}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) on \(\overrightarrow{\mathbf{a}}\) is
87755
If the vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{k}\) are the sides of a triangle \(\mathrm{ABC}\), then the length of the median through \(\mathrm{A}\) is
1 \(\sqrt{45}\)
2 \(\sqrt{18}\)
3 \(\sqrt{72}\)
4 \(\sqrt{33}\)
Explanation:
(D) : Given that, \(\overrightarrow{\mathrm{AB}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}\) \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) We know that, \(\overline{\mathrm{AM}}=\frac{1}{2}(\overline{\mathrm{AB}}+\overline{\mathrm{AC}})\) \(=\frac{1}{2}[(3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}})+(5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})]=\frac{8 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}}{2}\) \(=4 \hat{\mathrm{i}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) \(\therefore \mid \overline{\mathrm{AM}}=\sqrt{4^2+1^2+4^2}=\sqrt{33}\) Hence, the length of median is \(\sqrt{33}\).
COMEDK-2013
Vector Algebra
87756
If \(\vec{a}, \vec{b}, \vec{c}\) are three non-zero vectors such that each one of them are perpendicular to the sum of the other two vectors, then the value of \(|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}|^2\) is
87753
The projection of \(\vec{a}=2 \hat{i}+3 \hat{j}-2 \hat{k}\) on \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) is :
87754
A vector \(\vec{a}\) makes equal acute angles on the coordinate axis. Then the projection of vector \(\overrightarrow{\mathrm{b}}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) on \(\overrightarrow{\mathbf{a}}\) is
87755
If the vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{k}\) are the sides of a triangle \(\mathrm{ABC}\), then the length of the median through \(\mathrm{A}\) is
1 \(\sqrt{45}\)
2 \(\sqrt{18}\)
3 \(\sqrt{72}\)
4 \(\sqrt{33}\)
Explanation:
(D) : Given that, \(\overrightarrow{\mathrm{AB}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}\) \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) We know that, \(\overline{\mathrm{AM}}=\frac{1}{2}(\overline{\mathrm{AB}}+\overline{\mathrm{AC}})\) \(=\frac{1}{2}[(3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}})+(5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})]=\frac{8 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}}{2}\) \(=4 \hat{\mathrm{i}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) \(\therefore \mid \overline{\mathrm{AM}}=\sqrt{4^2+1^2+4^2}=\sqrt{33}\) Hence, the length of median is \(\sqrt{33}\).
COMEDK-2013
Vector Algebra
87756
If \(\vec{a}, \vec{b}, \vec{c}\) are three non-zero vectors such that each one of them are perpendicular to the sum of the other two vectors, then the value of \(|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}|^2\) is
87753
The projection of \(\vec{a}=2 \hat{i}+3 \hat{j}-2 \hat{k}\) on \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) is :
87754
A vector \(\vec{a}\) makes equal acute angles on the coordinate axis. Then the projection of vector \(\overrightarrow{\mathrm{b}}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) on \(\overrightarrow{\mathbf{a}}\) is
87755
If the vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{k}\) are the sides of a triangle \(\mathrm{ABC}\), then the length of the median through \(\mathrm{A}\) is
1 \(\sqrt{45}\)
2 \(\sqrt{18}\)
3 \(\sqrt{72}\)
4 \(\sqrt{33}\)
Explanation:
(D) : Given that, \(\overrightarrow{\mathrm{AB}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}\) \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) We know that, \(\overline{\mathrm{AM}}=\frac{1}{2}(\overline{\mathrm{AB}}+\overline{\mathrm{AC}})\) \(=\frac{1}{2}[(3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}})+(5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})]=\frac{8 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}}{2}\) \(=4 \hat{\mathrm{i}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) \(\therefore \mid \overline{\mathrm{AM}}=\sqrt{4^2+1^2+4^2}=\sqrt{33}\) Hence, the length of median is \(\sqrt{33}\).
COMEDK-2013
Vector Algebra
87756
If \(\vec{a}, \vec{b}, \vec{c}\) are three non-zero vectors such that each one of them are perpendicular to the sum of the other two vectors, then the value of \(|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}|^2\) is
87753
The projection of \(\vec{a}=2 \hat{i}+3 \hat{j}-2 \hat{k}\) on \(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) is :
87754
A vector \(\vec{a}\) makes equal acute angles on the coordinate axis. Then the projection of vector \(\overrightarrow{\mathrm{b}}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) on \(\overrightarrow{\mathbf{a}}\) is
87755
If the vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{k}\) are the sides of a triangle \(\mathrm{ABC}\), then the length of the median through \(\mathrm{A}\) is
1 \(\sqrt{45}\)
2 \(\sqrt{18}\)
3 \(\sqrt{72}\)
4 \(\sqrt{33}\)
Explanation:
(D) : Given that, \(\overrightarrow{\mathrm{AB}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}\) \(\overrightarrow{\mathrm{AC}}=5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) We know that, \(\overline{\mathrm{AM}}=\frac{1}{2}(\overline{\mathrm{AB}}+\overline{\mathrm{AC}})\) \(=\frac{1}{2}[(3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}})+(5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})]=\frac{8 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}}{2}\) \(=4 \hat{\mathrm{i}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) \(\therefore \mid \overline{\mathrm{AM}}=\sqrt{4^2+1^2+4^2}=\sqrt{33}\) Hence, the length of median is \(\sqrt{33}\).
COMEDK-2013
Vector Algebra
87756
If \(\vec{a}, \vec{b}, \vec{c}\) are three non-zero vectors such that each one of them are perpendicular to the sum of the other two vectors, then the value of \(|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}|^2\) is
