The scalar and vector product of two vectors, $\overrightarrow{\mathrm{a}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\left(3\hat{\mathrm{i}}-4\hat{\mathrm{j}}+5\hat{\mathrm{k}}\right)$ and $\overrightarrow{\mathrm{b}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\left(-2\hat{\mathrm{i}}+\hat{\mathrm{j}}-3\hat{\mathrm{k}}\right)$ is equal to:

1. \(-25\) & $\left(7\hat{\mathrm{i}}-\hat{\mathrm{j}}-5\hat{\mathrm{k}}\right)$

2. \(25\) & $\left(-7\hat{\mathrm{i}}+\hat{\mathrm{j}}-5\hat{\mathrm{k}}\right)$

3. \(0\) & $\left(-7\hat{\mathrm{i}}+\hat{\mathrm{j}}+3\hat{\mathrm{k}}\right)$

4. \(-25\) & $\left(-7\hat{\mathrm{i}}+\hat{\mathrm{j}}+5\hat{\mathrm{k}}\right)$

When vector $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\sqrt{3}\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>3\hat{\mathrm{j}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>2\hat{\mathrm{k}}$ intersects the Y-axis, it forms an angle of:

1. ${\sin }^{-1}\left(\frac{3}{\sqrt{14}}\right)$

2. ${\sin }^{-1}\left(\frac{\sqrt{7}}{4}\right)$

3. ${\cos }^{-1}\left(\frac{4}{3}\right)$

4. ${\cos }^{-1}\left(\frac{3}{5}\right)$

$\begin{array}{l}\text{\hspace{0.17em}If\hspace{0.17em}}|\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}}|=\sqrt{3}\overrightarrow{\mathrm{A}}\cdot \overrightarrow{\mathrm{B}}\text{, then the value of\hspace{0.17em}}\\ |\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}|\text{\hspace{0.17em}is:\hspace{0.17em}}\end{array}$

1. ${({\mathrm{A}}^2+{\mathrm{B}}^2+\frac{\mathrm{AB}}{\sqrt{3}})}^{1/2}$

2. A+B

3. ${({\mathrm{A}}^2+{\mathrm{B}}^2+\sqrt{3}\mathrm{AB})}^{1/2}$

4. ${({\mathrm{A}}^2+{\mathrm{B}}^2+\mathrm{AB})}^{1/2}$

If the angle between the vector $\overrightarrow{\mathrm{A}}\text{\hspace{0.17em}and\hspace{0.17em}}\overrightarrow{\mathrm{B}}$ is \(\theta\), the value of the product $(\overrightarrow{\mathrm{B}}\times \overrightarrow{\mathrm{A}})\cdot \overrightarrow{\mathrm{A}}$ is equal to:

1. zero

2. BA²\(\sin\theta \cos \theta\)

3. BA²\(\cos\theta\)

4. BA²\(\sin\theta\)

The component of a vector \(r\) along the \(X\)-axis will have the maximum value if:

Figure shows the orientation of two vectors \(u\) and \(v\) in the XY plane.
 If \(u = a\hat i + b\hat j\) and \(v = p\hat i + q\hat j\)

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Which of the following is correct?

Which of the following is not possible?
1. sin \(\theta\) = \(3\over5\)
2. sec \(\theta\) = \(100\)
3. cosec \(\theta\) = \(0.14\)
4. None of the above

If $∆ABC$ is right-angled at C, then the value of cos (A + B) is:

                               
1. \(0\)
2. \(1\)
3. \(\frac{1}{2}\)
4. \(\frac{\sqrt{3}}{2}\)

The incorrect statement/s is/are:
1. (b), (d)
2. (a), (c)
3. (b), (c), (d)
4. (a), (b)

The vector sum of two forces is perpendicular to their vector difference. In that case, the forces: