121170
If \(P(3,4,5), Q(4,6,3), R(-1,2,4), S(1,0,5)\), then the projection of \(R S\) on \(P Q\) is
1 \(-\frac{2}{3}\)
2 \(-\frac{4}{3}\)
3 \(\frac{1}{2}\)
4 2
Explanation:
B \(\mathrm{P}(3,4,5), \mathrm{Q}(4,6,3), \mathrm{R}(-1,2,4), \mathrm{S}(1,0,5)\) \(\text { Direction Ratio of } \mathrm{PQ}=[(4-3),(6-4),(3-5)]\)
\(=[1,2,-2]\)
\(\text { Direction Ratio of RS }=(2,-2,1)\)
\(\text { We know that, projection of } \vec{a} \text { on } \vec{b} \text { is given by } \frac{\vec{a} \cdot \vec{b}}{ \vert\vec{b} \vert}\)
\(=\frac{(\mathrm{RS}) \cdot(\mathrm{PQ})}{ \vert\mathrm{PQ} \vert}\)
\(=\frac{(2 \hat{i}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \cdot(\hat{i}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})}{\sqrt{(1)^2+(2)^2+(-2)^2}}\)
\(=\frac{2-4-2}{\sqrt{9}}=-\frac{4}{3}\)
\(\therefore\) The projection of RS on PQ is
\(=\frac{(\mathrm{RS}) \cdot(\mathrm{PQ})}{ \vert\mathrm{PQ} \vert}\)
\(=\frac{(2 \hat{i}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \cdot(\hat{i}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})}{\sqrt{(1)^2+(2)^2+(-2)^2}}\)
\(=\frac{2-4-2}{\sqrt{9}}=-\frac{4}{3}\)
Manipal UGET-2019
Three Dimensional Geometry
121172
If \(\left(\frac{1}{2}, \frac{1}{3}, \mathbf{n}\right)\) are the direction cosines of a line, then the value of \(n\) is
1 \(\frac{\sqrt{23}}{6}\)
2 \(\frac{23}{36}\)
3 \(\frac{2}{3}\)
4 \(\frac{3}{2}\)
Explanation:
A Since, \(\left(\frac{1}{2}, \frac{1}{3}, \mathrm{n}\right)\) are the direction cosines of a line, then \(\left(\frac{1}{2}\right)^2+\left(\frac{1}{3}\right)^2+\mathrm{n}^2=1\) \(\frac{1}{4}+\frac{1}{9}+\mathrm{n}^2=1\) \(\mathrm{n}^2=1-\frac{1}{4}-\frac{1}{9}\) \(\mathrm{n}^2=\frac{23}{36}\) \(\mathrm{n}=\sqrt{\frac{23}{36}}=\frac{\sqrt{23}}{6}\)Hence, option (a) is correct
Manipal UGET-2018
Three Dimensional Geometry
121173
A ray makes angles \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(Y\) and \(Z\) axes respectively. Then the value of the sine of the angle made by the ray with \(\mathrm{X}\)-axis is
1 \(\sqrt{3} / 2\)
2 \(1 / 2\)
3 \(1 / \sqrt{2}\)
4 1
Explanation:
A Given that, A ray make on angle \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(\mathrm{y}\) and \(\mathrm{z}\) axis respectively. Then we have, \(\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1\) \(\cos ^2 \alpha+\cos ^2(\pi / 3)+\cos ^2(\pi / 4)=1\) \(\cos ^2 \alpha+\left(\frac{1}{2}\right)^2+\left(\frac{1}{\sqrt{2}}\right)^2=1\) \(\cos ^2 \alpha+\frac{1}{4}+\frac{1}{2}=1\) \(\cos ^2 \alpha=1-\frac{1}{4}-\frac{1}{2}\) \(=\frac{4-1-2}{4}=\frac{1}{4}\) \(\because \quad \sin ^2 \alpha+\cos ^2 \alpha=1\) \(\therefore \quad \sin ^2 \alpha=1-\frac{1}{4}=\frac{3}{4}\) \(\sin \alpha=\frac{\sqrt{3}}{2}\)
AP EAMCET-07.07.2022
Three Dimensional Geometry
121174
If a line makes angles \(90^{\circ}, \mathbf{1 3 5}^{\circ}\) and \(45^{\circ}\) with the positive directions of \(x, y, z\)-axes respectively, then its direction cosines are......
121170
If \(P(3,4,5), Q(4,6,3), R(-1,2,4), S(1,0,5)\), then the projection of \(R S\) on \(P Q\) is
1 \(-\frac{2}{3}\)
2 \(-\frac{4}{3}\)
3 \(\frac{1}{2}\)
4 2
Explanation:
B \(\mathrm{P}(3,4,5), \mathrm{Q}(4,6,3), \mathrm{R}(-1,2,4), \mathrm{S}(1,0,5)\) \(\text { Direction Ratio of } \mathrm{PQ}=[(4-3),(6-4),(3-5)]\)
\(=[1,2,-2]\)
\(\text { Direction Ratio of RS }=(2,-2,1)\)
\(\text { We know that, projection of } \vec{a} \text { on } \vec{b} \text { is given by } \frac{\vec{a} \cdot \vec{b}}{ \vert\vec{b} \vert}\)
\(=\frac{(\mathrm{RS}) \cdot(\mathrm{PQ})}{ \vert\mathrm{PQ} \vert}\)
\(=\frac{(2 \hat{i}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \cdot(\hat{i}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})}{\sqrt{(1)^2+(2)^2+(-2)^2}}\)
\(=\frac{2-4-2}{\sqrt{9}}=-\frac{4}{3}\)
\(\therefore\) The projection of RS on PQ is
\(=\frac{(\mathrm{RS}) \cdot(\mathrm{PQ})}{ \vert\mathrm{PQ} \vert}\)
\(=\frac{(2 \hat{i}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \cdot(\hat{i}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})}{\sqrt{(1)^2+(2)^2+(-2)^2}}\)
\(=\frac{2-4-2}{\sqrt{9}}=-\frac{4}{3}\)
Manipal UGET-2019
Three Dimensional Geometry
121172
If \(\left(\frac{1}{2}, \frac{1}{3}, \mathbf{n}\right)\) are the direction cosines of a line, then the value of \(n\) is
1 \(\frac{\sqrt{23}}{6}\)
2 \(\frac{23}{36}\)
3 \(\frac{2}{3}\)
4 \(\frac{3}{2}\)
Explanation:
A Since, \(\left(\frac{1}{2}, \frac{1}{3}, \mathrm{n}\right)\) are the direction cosines of a line, then \(\left(\frac{1}{2}\right)^2+\left(\frac{1}{3}\right)^2+\mathrm{n}^2=1\) \(\frac{1}{4}+\frac{1}{9}+\mathrm{n}^2=1\) \(\mathrm{n}^2=1-\frac{1}{4}-\frac{1}{9}\) \(\mathrm{n}^2=\frac{23}{36}\) \(\mathrm{n}=\sqrt{\frac{23}{36}}=\frac{\sqrt{23}}{6}\)Hence, option (a) is correct
Manipal UGET-2018
Three Dimensional Geometry
121173
A ray makes angles \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(Y\) and \(Z\) axes respectively. Then the value of the sine of the angle made by the ray with \(\mathrm{X}\)-axis is
1 \(\sqrt{3} / 2\)
2 \(1 / 2\)
3 \(1 / \sqrt{2}\)
4 1
Explanation:
A Given that, A ray make on angle \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(\mathrm{y}\) and \(\mathrm{z}\) axis respectively. Then we have, \(\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1\) \(\cos ^2 \alpha+\cos ^2(\pi / 3)+\cos ^2(\pi / 4)=1\) \(\cos ^2 \alpha+\left(\frac{1}{2}\right)^2+\left(\frac{1}{\sqrt{2}}\right)^2=1\) \(\cos ^2 \alpha+\frac{1}{4}+\frac{1}{2}=1\) \(\cos ^2 \alpha=1-\frac{1}{4}-\frac{1}{2}\) \(=\frac{4-1-2}{4}=\frac{1}{4}\) \(\because \quad \sin ^2 \alpha+\cos ^2 \alpha=1\) \(\therefore \quad \sin ^2 \alpha=1-\frac{1}{4}=\frac{3}{4}\) \(\sin \alpha=\frac{\sqrt{3}}{2}\)
AP EAMCET-07.07.2022
Three Dimensional Geometry
121174
If a line makes angles \(90^{\circ}, \mathbf{1 3 5}^{\circ}\) and \(45^{\circ}\) with the positive directions of \(x, y, z\)-axes respectively, then its direction cosines are......
121170
If \(P(3,4,5), Q(4,6,3), R(-1,2,4), S(1,0,5)\), then the projection of \(R S\) on \(P Q\) is
1 \(-\frac{2}{3}\)
2 \(-\frac{4}{3}\)
3 \(\frac{1}{2}\)
4 2
Explanation:
B \(\mathrm{P}(3,4,5), \mathrm{Q}(4,6,3), \mathrm{R}(-1,2,4), \mathrm{S}(1,0,5)\) \(\text { Direction Ratio of } \mathrm{PQ}=[(4-3),(6-4),(3-5)]\)
\(=[1,2,-2]\)
\(\text { Direction Ratio of RS }=(2,-2,1)\)
\(\text { We know that, projection of } \vec{a} \text { on } \vec{b} \text { is given by } \frac{\vec{a} \cdot \vec{b}}{ \vert\vec{b} \vert}\)
\(=\frac{(\mathrm{RS}) \cdot(\mathrm{PQ})}{ \vert\mathrm{PQ} \vert}\)
\(=\frac{(2 \hat{i}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \cdot(\hat{i}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})}{\sqrt{(1)^2+(2)^2+(-2)^2}}\)
\(=\frac{2-4-2}{\sqrt{9}}=-\frac{4}{3}\)
\(\therefore\) The projection of RS on PQ is
\(=\frac{(\mathrm{RS}) \cdot(\mathrm{PQ})}{ \vert\mathrm{PQ} \vert}\)
\(=\frac{(2 \hat{i}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \cdot(\hat{i}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})}{\sqrt{(1)^2+(2)^2+(-2)^2}}\)
\(=\frac{2-4-2}{\sqrt{9}}=-\frac{4}{3}\)
Manipal UGET-2019
Three Dimensional Geometry
121172
If \(\left(\frac{1}{2}, \frac{1}{3}, \mathbf{n}\right)\) are the direction cosines of a line, then the value of \(n\) is
1 \(\frac{\sqrt{23}}{6}\)
2 \(\frac{23}{36}\)
3 \(\frac{2}{3}\)
4 \(\frac{3}{2}\)
Explanation:
A Since, \(\left(\frac{1}{2}, \frac{1}{3}, \mathrm{n}\right)\) are the direction cosines of a line, then \(\left(\frac{1}{2}\right)^2+\left(\frac{1}{3}\right)^2+\mathrm{n}^2=1\) \(\frac{1}{4}+\frac{1}{9}+\mathrm{n}^2=1\) \(\mathrm{n}^2=1-\frac{1}{4}-\frac{1}{9}\) \(\mathrm{n}^2=\frac{23}{36}\) \(\mathrm{n}=\sqrt{\frac{23}{36}}=\frac{\sqrt{23}}{6}\)Hence, option (a) is correct
Manipal UGET-2018
Three Dimensional Geometry
121173
A ray makes angles \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(Y\) and \(Z\) axes respectively. Then the value of the sine of the angle made by the ray with \(\mathrm{X}\)-axis is
1 \(\sqrt{3} / 2\)
2 \(1 / 2\)
3 \(1 / \sqrt{2}\)
4 1
Explanation:
A Given that, A ray make on angle \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(\mathrm{y}\) and \(\mathrm{z}\) axis respectively. Then we have, \(\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1\) \(\cos ^2 \alpha+\cos ^2(\pi / 3)+\cos ^2(\pi / 4)=1\) \(\cos ^2 \alpha+\left(\frac{1}{2}\right)^2+\left(\frac{1}{\sqrt{2}}\right)^2=1\) \(\cos ^2 \alpha+\frac{1}{4}+\frac{1}{2}=1\) \(\cos ^2 \alpha=1-\frac{1}{4}-\frac{1}{2}\) \(=\frac{4-1-2}{4}=\frac{1}{4}\) \(\because \quad \sin ^2 \alpha+\cos ^2 \alpha=1\) \(\therefore \quad \sin ^2 \alpha=1-\frac{1}{4}=\frac{3}{4}\) \(\sin \alpha=\frac{\sqrt{3}}{2}\)
AP EAMCET-07.07.2022
Three Dimensional Geometry
121174
If a line makes angles \(90^{\circ}, \mathbf{1 3 5}^{\circ}\) and \(45^{\circ}\) with the positive directions of \(x, y, z\)-axes respectively, then its direction cosines are......
121170
If \(P(3,4,5), Q(4,6,3), R(-1,2,4), S(1,0,5)\), then the projection of \(R S\) on \(P Q\) is
1 \(-\frac{2}{3}\)
2 \(-\frac{4}{3}\)
3 \(\frac{1}{2}\)
4 2
Explanation:
B \(\mathrm{P}(3,4,5), \mathrm{Q}(4,6,3), \mathrm{R}(-1,2,4), \mathrm{S}(1,0,5)\) \(\text { Direction Ratio of } \mathrm{PQ}=[(4-3),(6-4),(3-5)]\)
\(=[1,2,-2]\)
\(\text { Direction Ratio of RS }=(2,-2,1)\)
\(\text { We know that, projection of } \vec{a} \text { on } \vec{b} \text { is given by } \frac{\vec{a} \cdot \vec{b}}{ \vert\vec{b} \vert}\)
\(=\frac{(\mathrm{RS}) \cdot(\mathrm{PQ})}{ \vert\mathrm{PQ} \vert}\)
\(=\frac{(2 \hat{i}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \cdot(\hat{i}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})}{\sqrt{(1)^2+(2)^2+(-2)^2}}\)
\(=\frac{2-4-2}{\sqrt{9}}=-\frac{4}{3}\)
\(\therefore\) The projection of RS on PQ is
\(=\frac{(\mathrm{RS}) \cdot(\mathrm{PQ})}{ \vert\mathrm{PQ} \vert}\)
\(=\frac{(2 \hat{i}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \cdot(\hat{i}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})}{\sqrt{(1)^2+(2)^2+(-2)^2}}\)
\(=\frac{2-4-2}{\sqrt{9}}=-\frac{4}{3}\)
Manipal UGET-2019
Three Dimensional Geometry
121172
If \(\left(\frac{1}{2}, \frac{1}{3}, \mathbf{n}\right)\) are the direction cosines of a line, then the value of \(n\) is
1 \(\frac{\sqrt{23}}{6}\)
2 \(\frac{23}{36}\)
3 \(\frac{2}{3}\)
4 \(\frac{3}{2}\)
Explanation:
A Since, \(\left(\frac{1}{2}, \frac{1}{3}, \mathrm{n}\right)\) are the direction cosines of a line, then \(\left(\frac{1}{2}\right)^2+\left(\frac{1}{3}\right)^2+\mathrm{n}^2=1\) \(\frac{1}{4}+\frac{1}{9}+\mathrm{n}^2=1\) \(\mathrm{n}^2=1-\frac{1}{4}-\frac{1}{9}\) \(\mathrm{n}^2=\frac{23}{36}\) \(\mathrm{n}=\sqrt{\frac{23}{36}}=\frac{\sqrt{23}}{6}\)Hence, option (a) is correct
Manipal UGET-2018
Three Dimensional Geometry
121173
A ray makes angles \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(Y\) and \(Z\) axes respectively. Then the value of the sine of the angle made by the ray with \(\mathrm{X}\)-axis is
1 \(\sqrt{3} / 2\)
2 \(1 / 2\)
3 \(1 / \sqrt{2}\)
4 1
Explanation:
A Given that, A ray make on angle \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(\mathrm{y}\) and \(\mathrm{z}\) axis respectively. Then we have, \(\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1\) \(\cos ^2 \alpha+\cos ^2(\pi / 3)+\cos ^2(\pi / 4)=1\) \(\cos ^2 \alpha+\left(\frac{1}{2}\right)^2+\left(\frac{1}{\sqrt{2}}\right)^2=1\) \(\cos ^2 \alpha+\frac{1}{4}+\frac{1}{2}=1\) \(\cos ^2 \alpha=1-\frac{1}{4}-\frac{1}{2}\) \(=\frac{4-1-2}{4}=\frac{1}{4}\) \(\because \quad \sin ^2 \alpha+\cos ^2 \alpha=1\) \(\therefore \quad \sin ^2 \alpha=1-\frac{1}{4}=\frac{3}{4}\) \(\sin \alpha=\frac{\sqrt{3}}{2}\)
AP EAMCET-07.07.2022
Three Dimensional Geometry
121174
If a line makes angles \(90^{\circ}, \mathbf{1 3 5}^{\circ}\) and \(45^{\circ}\) with the positive directions of \(x, y, z\)-axes respectively, then its direction cosines are......
121170
If \(P(3,4,5), Q(4,6,3), R(-1,2,4), S(1,0,5)\), then the projection of \(R S\) on \(P Q\) is
1 \(-\frac{2}{3}\)
2 \(-\frac{4}{3}\)
3 \(\frac{1}{2}\)
4 2
Explanation:
B \(\mathrm{P}(3,4,5), \mathrm{Q}(4,6,3), \mathrm{R}(-1,2,4), \mathrm{S}(1,0,5)\) \(\text { Direction Ratio of } \mathrm{PQ}=[(4-3),(6-4),(3-5)]\)
\(=[1,2,-2]\)
\(\text { Direction Ratio of RS }=(2,-2,1)\)
\(\text { We know that, projection of } \vec{a} \text { on } \vec{b} \text { is given by } \frac{\vec{a} \cdot \vec{b}}{ \vert\vec{b} \vert}\)
\(=\frac{(\mathrm{RS}) \cdot(\mathrm{PQ})}{ \vert\mathrm{PQ} \vert}\)
\(=\frac{(2 \hat{i}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \cdot(\hat{i}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})}{\sqrt{(1)^2+(2)^2+(-2)^2}}\)
\(=\frac{2-4-2}{\sqrt{9}}=-\frac{4}{3}\)
\(\therefore\) The projection of RS on PQ is
\(=\frac{(\mathrm{RS}) \cdot(\mathrm{PQ})}{ \vert\mathrm{PQ} \vert}\)
\(=\frac{(2 \hat{i}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \cdot(\hat{i}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})}{\sqrt{(1)^2+(2)^2+(-2)^2}}\)
\(=\frac{2-4-2}{\sqrt{9}}=-\frac{4}{3}\)
Manipal UGET-2019
Three Dimensional Geometry
121172
If \(\left(\frac{1}{2}, \frac{1}{3}, \mathbf{n}\right)\) are the direction cosines of a line, then the value of \(n\) is
1 \(\frac{\sqrt{23}}{6}\)
2 \(\frac{23}{36}\)
3 \(\frac{2}{3}\)
4 \(\frac{3}{2}\)
Explanation:
A Since, \(\left(\frac{1}{2}, \frac{1}{3}, \mathrm{n}\right)\) are the direction cosines of a line, then \(\left(\frac{1}{2}\right)^2+\left(\frac{1}{3}\right)^2+\mathrm{n}^2=1\) \(\frac{1}{4}+\frac{1}{9}+\mathrm{n}^2=1\) \(\mathrm{n}^2=1-\frac{1}{4}-\frac{1}{9}\) \(\mathrm{n}^2=\frac{23}{36}\) \(\mathrm{n}=\sqrt{\frac{23}{36}}=\frac{\sqrt{23}}{6}\)Hence, option (a) is correct
Manipal UGET-2018
Three Dimensional Geometry
121173
A ray makes angles \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(Y\) and \(Z\) axes respectively. Then the value of the sine of the angle made by the ray with \(\mathrm{X}\)-axis is
1 \(\sqrt{3} / 2\)
2 \(1 / 2\)
3 \(1 / \sqrt{2}\)
4 1
Explanation:
A Given that, A ray make on angle \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(\mathrm{y}\) and \(\mathrm{z}\) axis respectively. Then we have, \(\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1\) \(\cos ^2 \alpha+\cos ^2(\pi / 3)+\cos ^2(\pi / 4)=1\) \(\cos ^2 \alpha+\left(\frac{1}{2}\right)^2+\left(\frac{1}{\sqrt{2}}\right)^2=1\) \(\cos ^2 \alpha+\frac{1}{4}+\frac{1}{2}=1\) \(\cos ^2 \alpha=1-\frac{1}{4}-\frac{1}{2}\) \(=\frac{4-1-2}{4}=\frac{1}{4}\) \(\because \quad \sin ^2 \alpha+\cos ^2 \alpha=1\) \(\therefore \quad \sin ^2 \alpha=1-\frac{1}{4}=\frac{3}{4}\) \(\sin \alpha=\frac{\sqrt{3}}{2}\)
AP EAMCET-07.07.2022
Three Dimensional Geometry
121174
If a line makes angles \(90^{\circ}, \mathbf{1 3 5}^{\circ}\) and \(45^{\circ}\) with the positive directions of \(x, y, z\)-axes respectively, then its direction cosines are......
