If \(\vec F=2\hat i+\hat j-\hat k\) and \(\vec r=3\hat i+2\hat j-2\hat k,\) then the scalar and vector products of \(\vec F\) and \(\vec r\) have the magnitudes, respectively, as:
1. \(5, ~\sqrt3\)
2. \(4,~ \sqrt5\)
3. \(10, ~\sqrt2\)
4. \(10,~2\)

The scalar and vector product of two vectors, $\overrightarrow{\mathrm{a}} = \left(3\hat{\mathrm{i}}-4\hat{\mathrm{j}}+5\hat{\mathrm{k}}\right)$ and $\overrightarrow{\mathrm{b}} = \left(-2\hat{\mathrm{i}}+\hat{\mathrm{j}}-3\hat{\mathrm{k}}\right)$ 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. \(-25\) & $\left(-7\hat{\mathrm{i}}+\hat{\mathrm{j}}+5\hat{\mathrm{k}}\right)$

For two vectors \(\vec A\) and \(\vec B\), |\(\vec A\)+\(\vec B\)|=|\(\vec A\) - \(\vec B\)| is always true when:

It is found that \(|\vec{A}+\vec{B}|=|\vec{A}|\). This necessarily implies:

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

Consider the quantities of pressure, power, energy, impulse, gravitational potential, electric charge, temperature, and area. Out of these, the only vector quantities are:

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

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

      
Which of the following is correct?