I am teaching a "linear algebra for engineers" course, and am currently building my lectures on Gram-Schmidt, QR-decomposition and least squares equation solving.
$\bullet$ I can motivate Gram-Schmidt perfectly well. Geometrically, if I have a tilted plane in $\mathbb{R}^3$ and want to describe geometric figures in it, I want to work in orthonormal coordinates in order for lengths and angles to have their usual meaning. And, more abstractly, Gram-Schmidt leads to orthogonal polynomials, which come up in tons of PDE applications.
$\bullet$ I can motivate least-squares perfectly well, it is the first basic step in statistical modeling, and also key to solving linear equations with numerical errors.
But I don't really see why students need $QR$-decomposition or, more precisely, why they need the matrix $R$. Orthogonal projection is given by $Q Q^T$. Least squares solving can be presented as $R^{-1} Q^T$, but the formula $(A^T A)^{-1} A^T$ is also available. The latter is certainly nicer for hand computation; I'm not sure what is actually lurking inside computer algebra systems.
Why is it worth teaching the students to compute $R$, and to think about what it means?