Question

When a right handed rectangular Cartesian system oxyz is rotated about the z-axis through an angle $\frac{\pi}{4}$ in the counter-clockwise direction it is found that a vector$\overset{\rightarrow}{a}$ has the components $2\sqrt{2}$, $3\sqrt{2}$ and 4. The components of $\overset{\rightarrow}{a}$in the oxyz coordinate system are-

• A. 5, –1, 4
• B. 5, –1, 4$\sqrt{2}$
• C. –1, –5, 4$\sqrt{2}$
• D. none

Answer 1

Answer: (D)

none

Answer 2

If $\overset{\hat{}}{i}$, $\overset{\hat{}}{j}$, $\overset{\hat{}}{k}$are the unit vectors in the oxyz system and $\overset{\hat{}}{i}$1, $\overset{\hat{}}{j}$1, $\overset{\hat{}}{k}$1 are the unit vectors in the system o x¢ y¢ z¢ obtained after rotation, then

$\overset{\hat{}}{i}$1 = $\frac { 1 } { \sqrt { 2 } }$ $\overset{\hat{}}{i}$+$\frac{1}{\sqrt{2}}$ $\overset{\hat{}}{j}$

$\overset{\hat{}}{j}$1 = $\frac { 1 } { \sqrt { 2 } }$ $\overset{\hat{}}{i}$+$\frac { 1 } { \sqrt { 2 } }$ $\overset{\hat{}}{j}$ and $\overset{\hat{}}{k}$1 = $\overset{\hat{}}{k}$

= $3 \sqrt { 2 }$ $\overset{\hat{}}{j}$ 1 + 4$\overset{\hat{}}{k}$1

=$2 \sqrt { 2 }$ $\left( \frac{1}{\sqrt{2}}\overset{萀\frac{1}{\sqrt{2}}\overset{\hat{}}{j}}{i}() \right)$+ $3 \sqrt { 2 }$ $\left( –\frac{1}{\sqrt{2}}\overset{萀\frac{1}{\sqrt{2}}\overset{\hat{}}{j}}{i}() \right)$ + 4$\overset{\hat{}}{k}$

= –$\overset{\hat{}}{i}$+ 5$\overset{\hat{}}{j}$ + 4$\overset{\hat{}}{k}$

So that the components in the oxyz system are – 1, 5, 4