### Video Transcript

Find the measure of the direction
angles of the vector π
represented by the given figure, correct to one decimal
place.

We will begin by writing the vector
π
in terms of its three components. In the π₯-direction, we travel
eight centimeters. Therefore, the π₯-component of the
vector is eight. In the π¦-direction, we travel 19
centimeters. Therefore, the π¦-component of
vector π
is 19. We travel nine centimeters in the
π§- or π§-direction. Therefore, the π§-component is
nine. Vector π
is equal to eight, 19,
nine. The magnitude of vector π
is equal
to the square root of eight squared plus 19 squared plus nine squared as we know
that the magnitude of any vector is equal to the square root of the sum of the
squares of the individual components.

Eight squared plus 19 squared plus
nine squared is equal to 506. Therefore, the magnitude of vector
π
is the square root of 506. We are asked to calculate the
direction angles. These are usually labeled πΌ, π½,
and πΎ, where πΌ is the angle between the π₯-axis and the vector π
, π½ is the angle
between the π¦-axis and the vector π
, and, finally, πΎ is the angle between the
π§-axis and the vector π
.

From our knowledge of the direction
cosines, we know that πΌ is equal to the inverse cos of π
sub π₯ over the magnitude
of π
, π½ is equal to the inverse cos of π
sub π¦ over the magnitude of π
, and πΎ
is equal to the inverse cos of π
sub π§ over the magnitude of π
, where π
sub π₯,
π
sub π¦, and π
sub π§ are the three components of vector π
. We will now clear some space to
calculate these values.

Angle πΌ is equal to the inverse
cos of eight over the square root of 506. This is equal to 69.1671 and so
on. Rounding to one decimal place,
angle πΌ is equal to 69.2 degrees. π½ is equal to the inverse cos of
19 over the square root of 506. This is equal to 32.3652 and so
on. Rounding this to one decimal place
gives us 32.4 degrees. Angle πΎ is equal to the inverse
cos of nine over the square root of 506. This is equal to 66.4156 and so
on. To one decimal place, angle πΎ is
66.4 degrees. The angles πΌ, π½, and πΎ can also
be written as π sub π₯, π sub π¦, and π sub π§. In this question, theyβre equal to
69.2 degrees, 32.4 degrees, and 66.4 degrees, respectively.