The vector v has magnitude 17 and makes an angle of 105 with the positive x direction.
Express v in terms of components, with i and j being unit vectors pointing in the positive x and y directions.
v = 17(cos105, sin105)
= 17cos105 i - 17sin105 j
since 105 = 60+45 degrees, and thus one of the "special" angles, they probably want exact values
cos105 = cos(60+45)
= cos60cos45 - sin60sin45
= (1/2)(√2/2) - (√3/2)(√2/2)
= (√2 - √6)/4
sin105 = sin(60+45)
= sin60cos45 + cos60sin45
= (√3/2)(√2/2) + (1/2)(√2/2)
= (√6 + √2)/4
v = (17/4)(√2-√6) i - (17/4)(√2 + √6) j
To express vector v in terms of components, we can use trigonometry to find the x and y components of the vector.
The x component of v can be found using the formula:
vx = v * cos(theta)
Where:
v is the magnitude of vector v (17)
theta is the angle that vector v makes with the positive x direction (105 degrees)
vx = 17 * cos(105°)
Using a calculator, we find that cos(105°) ≈ -0.2588
vx ≈ 17 * -0.2588 ≈ -4.3976
So, the x component of vector v is approximately -4.3976.
Similarly, the y component of v can be found using the formula:
vy = v * sin(theta)
vy = 17 * sin(105°)
Using a calculator, we find that sin(105°) ≈ 0.9659
vy ≈ 17 * 0.9659 ≈ 16.4153
So, the y component of vector v is approximately 16.4153.
Therefore, vector v can be expressed in terms of components as:
v = -4.3976i + 16.4153j
To express a vector in terms of its components, we need to break down the vector into its x and y components.
Given that the vector v has a magnitude of 17 and an angle of 105 degrees with the positive x-axis, we can use trigonometry to find its x and y components.
The x-component (v_x) can be determined using the formula:
v_x = v * cos(θ)
where v is the magnitude of the vector and θ is the angle made with the positive x-axis.
Substituting the given values:
v_x = 17 * cos(105°)
To find the y-component (v_y), we use the formula:
v_y = v * sin(θ)
Substituting the given values:
v_y = 17 * sin(105°)
Therefore, the vector v can be expressed in terms of its components as:
v = v_x * i + v_y * j
Substituting the calculated values:
v = (17 * cos(105°)) * i + (17 * sin(105°)) * j
Simplifying further, if needed:
v ≈ (-4.29 * i) + (16.23 * j)