Animated Solution for Mathematics - Vector Algebra: The number of vectors of unit length perpendicular to vectors $\vec{a}=(1,1,0)$ and $\vec{b}=(0,1,1)$ is

Imagine you are standing in a room, looking at the corner where two walls meet the floor. You have two vectors, $\vec{a}=(1,1,0)$ and $\vec{b}=(0,1,1)$.

Vector $\vec{a}$ lies comfortably in the $xy$-plane, while vector $\vec{b}$ stretches across the $yz$-plane. Because these vectors are not parallel, they define a unique, flat surface—a plane—that slices through our 3D coordinate system.

Our mission is to find the number of unit vectors that are perpendicular to both of these paths. We are looking for the 'normal' to the plane they define, much like finding the direction of a flagpole standing perfectly upright on that slanted surface.

When we need a vector that is simultaneously perpendicular to two others, we use the Cross Product. The cross product $\vec{n}=\vec{a}\times \vec{b}$ is mathematically guaranteed to produce a vector that is orthogonal to both $\vec{a}$ and $\vec{b}$.

To compute this, we set up the following determinant:

Expanding this determinant, we calculate the components:

For the $\hat{i}$ component: $(1\cdot 1-0\cdot 1)=1$.

For the $\hat{j}$ component (applying the alternating sign rule): $-(1\cdot 1-0\cdot 0)=-1$.

For the $\hat{k}$ component: $(1\cdot 1-1\cdot 0)=1$.

Thus, our normal vector is $\vec{n}=1\hat{i}-1\hat{j}+1\hat{k}$.

Unit Vectors

The question asks for unit vectors, which must have a magnitude of exactly $1$. First, we calculate the magnitude of our normal vector $\vec{n}$:

To normalize this, we divide the vector by its magnitude:

Many students overlook the symmetry of 3D space. We have found one unit vector, $\hat{n}$, that points 'up' from our plane, but its negative counterpart, $-\hat{n}$, points in the exact opposite direction. Both vectors are perpendicular to the plane and both possess a magnitude of $1$.

Therefore, we have two distinct unit vectors:

In the geometry of 3D space, for any two non-parallel vectors, there are always exactly two unit vectors perpendicular to the plane they form. We have successfully navigated the geometry and accounted for the symmetry of space.

The final answer is 2.