"\\vec{v_a}=\\vec{v_p}+\\vec{v_w}"
where "\\vec{v_a}" - vector of the actual velocity of the plane; "\\vec{v_p}" - vector of the own velocity of the plane; "\\vec{v_w}" - vector of the wind velocity.
"v_a=\\sqrt{v_{ax}^2+v_{ay}^2}"
where "v_a" - absolute value of the actual velocity of the plane; "v_{ax}" - x-component of the "\\vec{v_a}"; "v_{ay}" - y-component of the "\\vec{v_a}".
"v_{ax}" - the algebraic sum of the x-components of "\\vec{v_p}" and "\\vec{v_w}"; "v_{ay}" - the algebraic sum of the y-components of "\\vec{v_p}" and "\\vec{v_w}".
"v_{ax}=v_{px}+v_{wx}=v_p \\cdot \\cos{\\alpha} + v_w \\cdot \\cos{\\beta}"
"v_{ay}=v_{py}+v_{wy}=v_p \\cdot \\sin{\\alpha} + v_w \\cdot \\sin{\\beta}"
where "v_p" and "v_w" - absolute values of the own velocity of the plane and wind velocity, respectively; "v_{px}" and "v_{wx}" - x-components of the "\\vec{v_p}" and "\\vec{v_w}", respectively; "v_{py}" and "v_{wy}" - y-components of the "\\vec{v_p}" and "\\vec{v_w}", respectively; "\\alpha" - angle of the plane rising; "\\beta" - angle of the wind direction.
"v_{ax}=400 \\cdot \\cos{45 \\degree} + 40 \\cdot \\cos{135 \\degree} = 400 \\cdot \\frac{\\sqrt{2}}{2} - 40 \\cdot \\frac{\\sqrt{2}}{2} \\approx 254.56 \\; mph"
"v_{ay}=400 \\cdot \\sin{45 \\degree} + 40 \\cdot \\sin{135 \\degree} = 400 \\cdot \\frac{\\sqrt{2}}{2} + 40 \\cdot \\frac{\\sqrt{2}}{2} \\approx 311.13 \\; mph"
"v_a=\\sqrt{254.56^2+311.13^2} \\approx 402.0 \\; mph"
"\\tan{\\gamma}=\\frac{v_{ay}}{v_{ax}}"
where "\\gamma" - angle of the actual direction of the plane.
"\\tan{\\gamma}=\\frac{311.13}{254.56} \\approx 1.222"
"\\gamma \\approx 50.7 \\degree"
Answer: actual velocity of the plane is 402.0 mph, actual direction of the plane is 50.7"\\degree".
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