Mathematics : Vector Algebra

If the volume of parallelopiped with conterminus edges $4\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + \hat{\mathrm{k}}, - \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} 3\hat{\mathrm{i}} + 9\hat{\mathrm{j}} + \mathrm{p}\hat{\mathrm{k}}$ is 34 cubic units, then p is equal

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$\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}} = \overrightarrow{\mathrm{a}}\left(2\hat{\mathrm{i}} + \hat{\mathrm{j}}\right) = \overrightarrow{\mathrm{a}}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right) = 1, \mathrm{then} \overrightarrow{\mathrm{a}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\hat{\mathrm{i}} - \hat{\mathrm{k}}$

• $\frac{1}{3}\left(3\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{1}{3}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{1}{3}\left(3\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{points} \mathrm{whose} \mathrm{position} \mathrm{vectors} \mathrm{are} \\ 2\hat{\mathrm{i}} + \hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}, 6\hat{\mathrm{i}} - \hat{\mathrm{j}} +\hspace{0.17em}2\hat{\mathrm{k}} \mathrm{and} 14\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + \mathrm{p}\hat{\mathrm{k}} \\ \mathrm{are} \mathrm{collinear}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{p} \mathrm{is} \end{gathered}$$

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$$\begin{gathered} \mathrm{The} \mathrm{ratio} \mathrm{in} \mathrm{which} \hat{\mathrm{i}} + 2\hat{\mathrm{j}}\hspace{0.17em}+ 3\hat{\mathrm{k}} \mathrm{divides} \mathrm{the} \mathrm{join} \mathrm{of} - 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} +\hspace{0.17em}5\hat{\mathrm{k}} \mathrm{and} \\ 7\hat{\mathrm{i}} - \hat{\mathrm{k}} \mathrm{is} \end{gathered}$$

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$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = + \mathrm{\lambda }\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \mathrm{the} \mathrm{orthogonal} \mathrm{projection} \mathrm{of} \\ \overrightarrow{\mathrm{b}} \mathrm{on} \overrightarrow{\mathrm{a}} \mathrm{is} \frac{4}{3}\left(\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}\right), \mathrm{then} \mathrm{\lambda } = ? \end{gathered}$$

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$$\begin{gathered} \mathrm{The} \mathrm{volume} (\mathrm{in} \mathrm{cubic} \mathrm{units}) \mathrm{of} \mathrm{the} \mathrm{tetrahedron} \mathrm{with} \mathrm{edges} \hat{\mathrm{i}} + \hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}, \\ \hat{\mathrm{i}} - \hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hspace{0.17em}\hat{\mathrm{k}} \mathrm{is} \end{gathered}$$

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• $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

$$\begin{gathered} \mathrm{Let} \overrightarrow{\mathrm{a}} = {\mathrm{a}}_1\hat{\mathrm{i}} +\hspace{0.17em}{\mathrm{a}}_2\hat{\mathrm{j}} + {\mathrm{a}}_3\hat{\mathrm{k}} \\ \mathrm{Assertion} \left(\mathrm{A}\right) : \mathrm{The} \mathrm{identity} \\ {\left|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right|}^2 +\hspace{0.17em}{\left|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}}\right|}^2 + {\left|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right|}^2 = 2{\left|\overrightarrow{\mathrm{a}}\right|}^2 \mathrm{holds} \mathrm{for} \\ \overrightarrow{\mathrm{a}} \\ \mathrm{Reason} \left(\mathrm{R}\right) : \overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}} = {\mathrm{a}}_3\hat{\mathrm{j}} - {\mathrm{a}}_2\hat{\mathrm{k}}, \\ \overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}} = {\mathrm{a}}_1\hat{\mathrm{k}} - {\mathrm{a}}_3\hat{\mathrm{i}}, \overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}} = {\mathrm{a}}_2\hat{\mathrm{i}} - {\mathrm{a}}_1\hat{\mathrm{j}} \\ \mathrm{Wich} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• Both (A) and (R) are true and (R) is the correct reason for (A)

• Both (A) and (R) are true but (R) is not the correct reason for (A)

• (A) is true, (R) is false

• A) is false, (R) is true

The position vectors of P and Q are $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ respectively. If R is a point on PQ such that $\overrightarrow{\mathrm{PR}} = 5\overrightarrow{\mathrm{PQ}}$, then the position vector of R is

• $5\overrightarrow{\mathrm{b}} - 4\overrightarrow{\mathrm{a}}$

• $5\overrightarrow{\mathrm{b}} + 4\overrightarrow{\mathrm{a}}$

• $4\overrightarrow{\mathrm{b}} - 5\overrightarrow{\mathrm{a}}$

• $4\overrightarrow{\mathrm{b}} + 5\overrightarrow{\mathrm{a}}$

If the points with position vectors $60\hat{\mathrm{i}} + 3\hat{\mathrm{j}}, 40\hat{\mathrm{i}} - 8\hat{\mathrm{j}} \mathrm{and} \mathrm{a}\hat{\mathrm{i}} - 52\hat{\mathrm{j}} \mathrm{are} \mathrm{collinear}, \mathrm{then} \mathrm{a} \mathrm{is} \mathrm{equal} \mathrm{to}$  are collinear,then a is equal to

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If the position vectors of A, B and C are respectively $2\hat{\mathrm{I}} - \hat{\mathrm{J}} + \hat{\mathrm{K}}, \hat{\mathrm{I}} - 3\hat{\mathrm{J}} - 5\hat{\mathrm{K}} \mathrm{and} 3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \mathrm{then} {\cos }^2\left(\mathrm{A}\right) = ?$ 

• 0

• $\frac{6}{41}$

• $\frac{35}{41}$

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