Given X=- i- j+ k, Y=- i+2 j+ k and Z= i+ k, which is TRUE?

X makes 30° with the normal to the plane containing Y and Z.

Z makes 60° with the normal to the plane containing X and Y.

GATE 2020 MT – Question 23

Engineering Mathematics · Calculus and vector calculus · 1 mark · Multiple choice

Given $\mathbf X=-\mathbf i-\mathbf j+\mathbf k$, $\mathbf Y=-\mathbf i+2\mathbf j+\mathbf k$ and $\mathbf Z=\mathbf i+\mathbf k$, which is TRUE?

X, Y and Z are mutually perpendicular.
X, Y and Z are coplanar.
X makes 30° with the normal to the plane containing Y and Z.
Z makes 60° with the normal to the plane containing X and Y.

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Correct answer: (A) X, Y and Z are mutually perpendicular.

Explanation

Calculate $X\cdot Y=1-2+1=0$, $X\cdot Z=-1+0+1=0$, and $Y\cdot Z=-1+0+1=0$. All are nonzero vectors, so they are pairwise perpendicular. Thus A. Their scalar triple product is nonzero, ruling out coplanarity; Z is normal to the X–Y plane, not at 60° to its normal.

GATE 2020 MT – Question 23

Explanation

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