Step 1: Translate the Perpendicularity Conditions into Equations
We are given that each vector is perpendicular to the sum of the remaining two vectors:
$$\vec{p} \cdot (\vec{q} + \vec{r}) = 0 \implies \vec{p} \cdot \vec{q} + \vec{p} \cdot \vec{r} = 0$$
$$\vec{q} \cdot (\vec{p} + \vec{r}) = 0 \implies \vec{q} \cdot \vec{p} + \vec{q} \cdot \vec{r} = 0$$
$$\vec{r} \cdot (\vec{p} + \vec{q}) = 0 \implies \vec{r} \cdot \vec{p} + \vec{r} \cdot \vec{q} = 0$$
Step 2: Sum the Equations
Add the three scalar equations together:
$$(\vec{p} \cdot \vec{q} + \vec{p} \cdot \vec{r}) + (\vec{q} \cdot \vec{p} + \vec{q} \cdot \vec{r}) + (\vec{r} \cdot \vec{p} + \vec{r} \cdot \vec{q}) = 0$$
Since the dot product is commutative ($$\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$$), we can collect like terms:
$$2(\vec{p} \cdot \vec{q} + \vec{q} \cdot \vec{r} + \vec{r} \cdot \vec{p}) = 0$$
Dividing by 2 yields:
$$\vec{p} \cdot \vec{q} + \vec{q} \cdot \vec{r} + \vec{r} \cdot \vec{p} = 0$$
Step 3: Use the Vector Magnitude Identity
We expand the square of the magnitude of the sum vector using the algebraic identity for vectors:
$$|\vec{p} + \vec{q} + \vec{r}|^2 = |\vec{p}|^2 + |\vec{q}|^2 + |\vec{r}|^2 + 2(\vec{p} \cdot \vec{q} + \vec{q} \cdot \vec{r} + \vec{r} \cdot \vec{p})$$
Step 4: Substitute Known Values
We are given the individual magnitudes:
$$|\vec{p}| = 2 \implies |\vec{p}|^2 = 4$$
$$|\vec{q}| = 3 \implies |\vec{q}|^2 = 9$$
$$|\vec{r}| = 4 \implies |\vec{r}|^2 = 16$$
From Step 2, we know that the cross-product/dot-product combination term is zero:
$$2(\vec{p} \cdot \vec{q} + \vec{q} \cdot \vec{r} + \vec{r} \cdot \vec{p}) = 0$$
Substitute these values into the expansion equation:
$$|\vec{p} + \vec{q} + \vec{r}|^2 = 4 + 9 + 16 + 0$$
$$|\vec{p} + \vec{q} + \vec{r}|^2 = 29$$
Step 5: Solve for the Final Magnitude
Taking the square root of both sides gives the absolute magnitude:
$$|\vec{p} + \vec{q} + \vec{r}| = \sqrt{29}$$
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