Question

Given that i - j, i + 2j are two vectors. Find a unit vector coplanar with these vectors and perpendicular to the first vector i - j. Find also the unit vector which is perpendicular to the plane of the two given vectors. Do you thus obtain an orthonormal triad ?

• A. $k$
• B. $\frac{1}{\sqrt{2}}(i+j);$
• C. $-k$
• D. $\frac{1}{\sqrt{2}}(i-j);$

Answer 1

Answer: (A), (B)

The correct options are
A $k$
B $\frac{1}{\sqrt{2}}(i+j);$

Consider the first vector $i-j$

Now a vector co-planar with the two vectors and perpendicular to $i-j$ will be of the form $r\frac{(i-j)}{\sqrt{2}}$ where r is a scalar quantity.

Since we require a unit vector, therefore r=1.

Now the normal vector of the plane containing the two specified vectors will be

$(i-j)\times (i+2j)$
$=2k+k$
$=3k$

Thus unit vector along the normal of the plane will be
$=k$.