Calculate the acute angle between 2x - 4y - z = -5 and 3x + 4y + 5z = -6

Let
${\bf A}$ = vector perpendicular to the first plane
${\bf B}$ = vector perpendicular to the second plane
 

${\bf A} = 2{\bf i} - 4{\bf j} - {\bf k}$

${\bf B} = 3{\bf i} + 4{\bf j} + 5{\bf k}$
 

The required angle between the two planes is equal to the angle between vectors $\mathbf{A}$ and $\mathbf{B}$. The formula for the angle between two vectors is given by

 

Dot product of $\mathbf{A}$ and $\mathbf{B}$
$\mathbf{A} \cdot \mathbf{B} = 2(3) - 4(4) - 1(5)$

$\mathbf{A} \cdot \mathbf{B} = -15$
 

Magnitude of vector $\mathbf{A}$
$\left\| \mathbf{A} \right\| = \sqrt{2^2 + (-4)^2 + (-1)^2}$

$\left\| \mathbf{A} \right\| = \sqrt{21}$
 

Magnitude of vector $\mathbf{B}$
$\left\| \mathbf{B} \right\| = \sqrt{3^2 + 4^2 + 5^2}$

$\left\| \mathbf{B} \right\| = 5\sqrt{2}$

 

Hence,
$\cos \theta = \dfrac{-15}{\sqrt{21} \left( 5\sqrt{2} \right)}$

$\theta = 117.56^\circ$
 

$\text{Acute angle} = 180^\circ - \theta$

$\text{Acute angle} = 180^\circ - 117.56^\circ$

$\text{Acute angle} = 62.42^\circ$   ←   answer