Question(d) Show that, if $\vec{a}$ and $\vec{b}$ are non-zero vectors belonging to $\mathbf{R}^{3}$ , then $\vec{a}$ and $\vec{b}$ can't be both orthogonal and parallel

Studdy Solution

STEP 1

What is this asking? Can two non-zero 3D vectors be both perpendicular and parallel? Watch out! Remember what parallel and orthogonal really mean!

STEP 2

1. Define parallel and orthogonal vectors
2. Explore the parallel case
3. Explore the orthogonal case
4. Combine and conclude

STEP 3

Alright, let's define what it means for two vectors to be parallel.
Two vectors $\vec{a}$ and $\vec{b}$ are parallel if one is a scalar multiple of the other.
That means $\vec{b} = k\cdot\vec{a}$ for some non-zero scalar $k$ .
Think of it like this: they point in the same direction (or exactly opposite directions), just maybe with different lengths!

STEP 4

Now, what does it mean for two vectors to be orthogonal?
It means they are perpendicular to each other, forming a right angle.
Mathematically, this means their dot product is zero: $\vec{a} \cdot \vec{b} = 0$ .

STEP 5

Let's imagine $\vec{a}$ and $\vec{b}$ are parallel.
So, $\vec{b} = k\cdot\vec{a}$ .
Let's write $\vec{a}$ as $\langle a_1, a_2, a_3 \rangle$ and $\vec{b}$ as $\langle b_1, b_2, b_3 \rangle$ .
Then, $\langle b_1, b_2, b_3 \rangle = k\cdot\langle a_1, a_2, a_3 \rangle = \langle k\cdot a_1, k\cdot a_2, k\cdot a_3 \rangle$ .

STEP 6

Now, let's also assume they're orthogonal.
This means their dot product is zero.
Let's substitute our expression for $\vec{b}$ from the parallel case: $\vec{a} \cdot \vec{b} = \vec{a} \cdot (k\cdot\vec{a}) = k\cdot(\vec{a} \cdot \vec{a}) = 0$ .

STEP 7

Remember that the dot product of a vector with itself is the square of its magnitude: $\vec{a} \cdot \vec{a} = ||\vec{a}||^2$ .
So, we have $k\cdot||\vec{a}||^2 = 0$ .

STEP 8

We're given that $\vec{a}$ is non-zero.
This means its magnitude, $||\vec{a}||$ , is also non-zero, and so is $||\vec{a}||^2$ .
The only way for $k\cdot||\vec{a}||^2 = 0$ to be true when $||\vec{a}||^2$ is not zero, is if $k = 0$ .

STEP 9

But wait!
If $k=0$ , then $\vec{b} = 0\cdot\vec{a} = \vec{0}$ , meaning $\vec{b}$ is the zero vector.
But we were told both vectors are non-zero!
We've reached a contradiction!

STEP 10

This means our initial assumption—that $\vec{a}$ and $\vec{b}$ are both parallel and orthogonal—must be false!
Boom!

STEP 11

Therefore, two non-zero vectors in $\mathbf{R}^3$ cannot be both parallel and orthogonal.

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