361785
Vector \({\vec A}\) makes equal angles with \(x\), \(y\) and \(z\) axis. Value of its components (in terms of magnitude of \({\vec A}\)) will be
1 \(\frac{A}{{\sqrt 2 }}\)
2 \(\frac{A}{{\sqrt 3 }}\)
3 \(\frac{{\sqrt 3 }}{A}\)
4 \(\sqrt 3 A\)
Explanation:
Let the component of \({\vec A}\) makes angles \(\alpha ,\,\beta \,{\rm{and}}\,\gamma \) with \(x\),\(y\) and \(z\) axis respectively then \(\alpha = \beta \, = \,\gamma \) \({\cos ^2}\alpha + {\cos ^2}\beta \, + {\cos ^2}\,\gamma = 1\) \( \Rightarrow 3{\cos ^2}\alpha = 1 \Rightarrow \cos \alpha = \frac{1}{{\sqrt 3 }}\) \(\therefore {A_x} = {A_y} = {A_z} = A\cos \alpha = \frac{A}{{\sqrt 3 }}\)
PHXI04:MOTION IN A PLANE
361786
The component of a vector \(r\) along \(X\)-axis will have maximum value if
1 \(r\) is along positive \(Y\)-axis
2 \(r\) is along positive \(X\)-axis
3 \(r\) makes an angle of \(45^\circ \) with the \(X\)-axis
4 \(r\) is along negative \(Y\)-axis
Explanation:
If \(r\) makes an angle \(\theta \) with \(x\)-axis, then component of \(r\) along \(x\)-axis.\( = r\cos \theta \) It will be maximum if \(\cos \theta = \max = 1 \Rightarrow \theta = {0^0},\) i.e., \(r\) is along positive \(x\)-axis.
NCERT Exemplar
PHXI04:MOTION IN A PLANE
361787
The \(X\) and \(Y\) components of a force \(F\) acting at \(30^\circ \) to \(x\)-axis are respectively
1 \(\frac{F}{2},\,\frac{{\sqrt 3 }}{2}F\)
2 \(\frac{F}{{\sqrt 2 }},{\rm{ }}F\)
3 \(F,{\rm{ }}\frac{F}{{\sqrt 2 }}\)
4 \(\frac{{\sqrt 3 }}{2}F,\,\frac{F}{2}\,\)
Explanation:
The \(X\) component of force \(F\) is \({F_x} = F\cos 30^\circ = F \times \frac{{\sqrt 3 }}{2} = \frac{{\sqrt 3 }}{2}F\) The \(Y\) component of force \(F\) is \({F_y} = F\sin 30^\circ = F \times \frac{1}{2} = \frac{1}{2}F\)
PHXI04:MOTION IN A PLANE
361788
If \(\vec{A}=3 \hat{i}+5 \hat{j}-7 \hat{k}\), the direction of cosines of the vector \(\vec{A}\) are:
NEET Test Series from KOTA - 10 Papers In MS WORD
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PHXI04:MOTION IN A PLANE
361785
Vector \({\vec A}\) makes equal angles with \(x\), \(y\) and \(z\) axis. Value of its components (in terms of magnitude of \({\vec A}\)) will be
1 \(\frac{A}{{\sqrt 2 }}\)
2 \(\frac{A}{{\sqrt 3 }}\)
3 \(\frac{{\sqrt 3 }}{A}\)
4 \(\sqrt 3 A\)
Explanation:
Let the component of \({\vec A}\) makes angles \(\alpha ,\,\beta \,{\rm{and}}\,\gamma \) with \(x\),\(y\) and \(z\) axis respectively then \(\alpha = \beta \, = \,\gamma \) \({\cos ^2}\alpha + {\cos ^2}\beta \, + {\cos ^2}\,\gamma = 1\) \( \Rightarrow 3{\cos ^2}\alpha = 1 \Rightarrow \cos \alpha = \frac{1}{{\sqrt 3 }}\) \(\therefore {A_x} = {A_y} = {A_z} = A\cos \alpha = \frac{A}{{\sqrt 3 }}\)
PHXI04:MOTION IN A PLANE
361786
The component of a vector \(r\) along \(X\)-axis will have maximum value if
1 \(r\) is along positive \(Y\)-axis
2 \(r\) is along positive \(X\)-axis
3 \(r\) makes an angle of \(45^\circ \) with the \(X\)-axis
4 \(r\) is along negative \(Y\)-axis
Explanation:
If \(r\) makes an angle \(\theta \) with \(x\)-axis, then component of \(r\) along \(x\)-axis.\( = r\cos \theta \) It will be maximum if \(\cos \theta = \max = 1 \Rightarrow \theta = {0^0},\) i.e., \(r\) is along positive \(x\)-axis.
NCERT Exemplar
PHXI04:MOTION IN A PLANE
361787
The \(X\) and \(Y\) components of a force \(F\) acting at \(30^\circ \) to \(x\)-axis are respectively
1 \(\frac{F}{2},\,\frac{{\sqrt 3 }}{2}F\)
2 \(\frac{F}{{\sqrt 2 }},{\rm{ }}F\)
3 \(F,{\rm{ }}\frac{F}{{\sqrt 2 }}\)
4 \(\frac{{\sqrt 3 }}{2}F,\,\frac{F}{2}\,\)
Explanation:
The \(X\) component of force \(F\) is \({F_x} = F\cos 30^\circ = F \times \frac{{\sqrt 3 }}{2} = \frac{{\sqrt 3 }}{2}F\) The \(Y\) component of force \(F\) is \({F_y} = F\sin 30^\circ = F \times \frac{1}{2} = \frac{1}{2}F\)
PHXI04:MOTION IN A PLANE
361788
If \(\vec{A}=3 \hat{i}+5 \hat{j}-7 \hat{k}\), the direction of cosines of the vector \(\vec{A}\) are:
361785
Vector \({\vec A}\) makes equal angles with \(x\), \(y\) and \(z\) axis. Value of its components (in terms of magnitude of \({\vec A}\)) will be
1 \(\frac{A}{{\sqrt 2 }}\)
2 \(\frac{A}{{\sqrt 3 }}\)
3 \(\frac{{\sqrt 3 }}{A}\)
4 \(\sqrt 3 A\)
Explanation:
Let the component of \({\vec A}\) makes angles \(\alpha ,\,\beta \,{\rm{and}}\,\gamma \) with \(x\),\(y\) and \(z\) axis respectively then \(\alpha = \beta \, = \,\gamma \) \({\cos ^2}\alpha + {\cos ^2}\beta \, + {\cos ^2}\,\gamma = 1\) \( \Rightarrow 3{\cos ^2}\alpha = 1 \Rightarrow \cos \alpha = \frac{1}{{\sqrt 3 }}\) \(\therefore {A_x} = {A_y} = {A_z} = A\cos \alpha = \frac{A}{{\sqrt 3 }}\)
PHXI04:MOTION IN A PLANE
361786
The component of a vector \(r\) along \(X\)-axis will have maximum value if
1 \(r\) is along positive \(Y\)-axis
2 \(r\) is along positive \(X\)-axis
3 \(r\) makes an angle of \(45^\circ \) with the \(X\)-axis
4 \(r\) is along negative \(Y\)-axis
Explanation:
If \(r\) makes an angle \(\theta \) with \(x\)-axis, then component of \(r\) along \(x\)-axis.\( = r\cos \theta \) It will be maximum if \(\cos \theta = \max = 1 \Rightarrow \theta = {0^0},\) i.e., \(r\) is along positive \(x\)-axis.
NCERT Exemplar
PHXI04:MOTION IN A PLANE
361787
The \(X\) and \(Y\) components of a force \(F\) acting at \(30^\circ \) to \(x\)-axis are respectively
1 \(\frac{F}{2},\,\frac{{\sqrt 3 }}{2}F\)
2 \(\frac{F}{{\sqrt 2 }},{\rm{ }}F\)
3 \(F,{\rm{ }}\frac{F}{{\sqrt 2 }}\)
4 \(\frac{{\sqrt 3 }}{2}F,\,\frac{F}{2}\,\)
Explanation:
The \(X\) component of force \(F\) is \({F_x} = F\cos 30^\circ = F \times \frac{{\sqrt 3 }}{2} = \frac{{\sqrt 3 }}{2}F\) The \(Y\) component of force \(F\) is \({F_y} = F\sin 30^\circ = F \times \frac{1}{2} = \frac{1}{2}F\)
PHXI04:MOTION IN A PLANE
361788
If \(\vec{A}=3 \hat{i}+5 \hat{j}-7 \hat{k}\), the direction of cosines of the vector \(\vec{A}\) are:
361785
Vector \({\vec A}\) makes equal angles with \(x\), \(y\) and \(z\) axis. Value of its components (in terms of magnitude of \({\vec A}\)) will be
1 \(\frac{A}{{\sqrt 2 }}\)
2 \(\frac{A}{{\sqrt 3 }}\)
3 \(\frac{{\sqrt 3 }}{A}\)
4 \(\sqrt 3 A\)
Explanation:
Let the component of \({\vec A}\) makes angles \(\alpha ,\,\beta \,{\rm{and}}\,\gamma \) with \(x\),\(y\) and \(z\) axis respectively then \(\alpha = \beta \, = \,\gamma \) \({\cos ^2}\alpha + {\cos ^2}\beta \, + {\cos ^2}\,\gamma = 1\) \( \Rightarrow 3{\cos ^2}\alpha = 1 \Rightarrow \cos \alpha = \frac{1}{{\sqrt 3 }}\) \(\therefore {A_x} = {A_y} = {A_z} = A\cos \alpha = \frac{A}{{\sqrt 3 }}\)
PHXI04:MOTION IN A PLANE
361786
The component of a vector \(r\) along \(X\)-axis will have maximum value if
1 \(r\) is along positive \(Y\)-axis
2 \(r\) is along positive \(X\)-axis
3 \(r\) makes an angle of \(45^\circ \) with the \(X\)-axis
4 \(r\) is along negative \(Y\)-axis
Explanation:
If \(r\) makes an angle \(\theta \) with \(x\)-axis, then component of \(r\) along \(x\)-axis.\( = r\cos \theta \) It will be maximum if \(\cos \theta = \max = 1 \Rightarrow \theta = {0^0},\) i.e., \(r\) is along positive \(x\)-axis.
NCERT Exemplar
PHXI04:MOTION IN A PLANE
361787
The \(X\) and \(Y\) components of a force \(F\) acting at \(30^\circ \) to \(x\)-axis are respectively
1 \(\frac{F}{2},\,\frac{{\sqrt 3 }}{2}F\)
2 \(\frac{F}{{\sqrt 2 }},{\rm{ }}F\)
3 \(F,{\rm{ }}\frac{F}{{\sqrt 2 }}\)
4 \(\frac{{\sqrt 3 }}{2}F,\,\frac{F}{2}\,\)
Explanation:
The \(X\) component of force \(F\) is \({F_x} = F\cos 30^\circ = F \times \frac{{\sqrt 3 }}{2} = \frac{{\sqrt 3 }}{2}F\) The \(Y\) component of force \(F\) is \({F_y} = F\sin 30^\circ = F \times \frac{1}{2} = \frac{1}{2}F\)
PHXI04:MOTION IN A PLANE
361788
If \(\vec{A}=3 \hat{i}+5 \hat{j}-7 \hat{k}\), the direction of cosines of the vector \(\vec{A}\) are:
