= Solution
Let $p:\mathbb P(E)\to X$ be the <projectivization of a real vector bundle>, let $L_E$ be its <tautological bundle>, and put
$$
u=w_1(L_E)\in H^1(\mathbb P(E);\mathbb F_2).
$$
The mod-two projective bundle formula says that $H^*(\mathbb P(E);\mathbb F_2)$ is a free $H^*(X;\mathbb F_2)$-module on $1,u,\ldots,u^{d-1}$. The <Projective bundle definition of Stiefel–Whitney classes> is the unique relation
$$
u^d+p^*w_1(E)u^{d-1}+\cdots+p^*w_d(E)=0.
$$
Apply the <splitting principle for real vector bundles>. After an injective pullback, write
$$
E=L_1\oplus\cdots\oplus L_d,
\qquad
E'=L'_1\oplus\cdots\oplus L'_{d'}.
$$
If $x_i=w_1(L_i)$, the projective-bundle relation factors as
$$
\prod_{i=1}^d(u+x_i)=0,
$$
so
$$
w(E)=\prod_{i=1}^d(1+x_i).
$$
The line summands of $E\oplus E'$ are the union of the two lists, hence
$$
w(E\oplus E')=w(E)w(E').
$$
Comparing the degree-$k$ components proves the <Whitney product formula for Stiefel–Whitney classes>
$$
w_k(E\oplus E')=\sum_{i+j=k}w_i(E)w_j(E').
$$
Injectivity of the splitting pullback returns the identity to $X$.
For real line bundles, the transition functions take values in $O(1)=\{\pm1\}$. Tensor product multiplies these signs, while the identification $\{\pm1\}\cong\mathbb Z/2$ turns multiplication into addition. The corresponding degree-one characteristic classes therefore satisfy the <First Stiefel–Whitney class of a tensor product of real line bundles> formula
$$
w_1(L\otimes L')=w_1(L)+w_1(L').
$$
Equivalently, this follows from the classification of real line bundles by $H^1(X;\mathbb F_2)$.
Now take $M=\mathbb{RP}^n$ and $E=k\gamma_{\mathbb R}^{1,n+1}$. Since
$$
E=\gamma_{\mathbb R}^{1,n+1}\otimes\mathbb R^k,
$$
a line in $E_x$ is the fixed line $\gamma_x$ tensored with a line in $\mathbb R^k$. Thus the <projectivization of copies of the real tautological line bundle> is
$$
\mathbb P(E)\cong\mathbb{RP}^n\times\mathbb{RP}^{k-1}.
$$
Let $x$ and $v$ be the degree-one generators pulled back from the first and second factors. The <mod-two cohomology ring of real projective space> and the <Künneth theorem> give
$$
H^*(\mathbb P(E);\mathbb F_2)
\cong
\mathbb F_2[x,v]/(x^{n+1},v^k).
$$
The tautological line $L_E$ is the tensor product of the two tautological lines, so $w_1(L_E)=x+v$. In the alternative generator $u=w_1(L_E)$, the same ring is
$$
\mathbb F_2[x,u]/(x^{n+1},(u+x)^k).
$$
The stable tangent-bundle identity
$$
T\mathbb{RP}^n\oplus\mathbf1\cong(n+1)\gamma_{\mathbb R}^{1,n+1}
$$
gives
$$
w(T\mathbb{RP}^n)=(1+x)^{n+1}.
$$
For the vertical part of the <tangent bundle of a projectivized real vector bundle>,
$$
\mathbf1\oplus\operatorname{Hom}(L_E,\omega_E)
\cong L_E^*\otimes p^*E.
$$
Each of the $k$ line summands on the right has first Stiefel–Whitney class
$$
w_1(L_E^*\otimes p^*\gamma)=w_1(L_E)+x=v.
$$
The <Whitney product formula for Stiefel–Whitney classes> therefore yields
$$
w(T\mathbb P(E))
=(1+x)^{n+1}(1+v)^k,
$$
the <Total Stiefel–Whitney class of the projectivization of copies of the tautological line>.
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