Write the vector v in the form ai+ bj, given its magnitude is [v] = 24 and the angle it makes with the positive x-axis is a = 300°. v= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type your answer in the form ai+bj.)

Calculate the x-component using a = ∣ v ∣ cos ( α ) , where ∣ v ∣ = 24 and α = 30 0 ∘ .
Calculate the y-component using b = ∣ v ∣ sin ( α ) , where ∣ v ∣ = 24 and α = 30 0 ∘ .
Determine that cos ( 30 0 ∘ ) = 2 1 ​ and sin ( 30 0 ∘ ) = − 2 3 ​ ​ .
Express the vector v in the form ai + bj : 12 i − 12 3 ​ j ​ .

Explanation

Problem Analysis We are given that the magnitude of the vector v is ∣ v ∣ = 24 and the angle it makes with the positive x-axis is α = 30 0 ∘ . We need to express the vector v in the form ai + bj , where a and b are the scalar components of the vector along the x and y axes, respectively.

Component Formulas To find the components a and b , we use the following formulas:


a = ∣ v ∣ cos ( α )
b = ∣ v ∣ sin ( α )

Substitution Substitute the given values into the formulas:

a = 24 cos ( 30 0 ∘ )
b = 24 sin ( 30 0 ∘ )

Trigonometric Values We know that 30 0 ∘ = 36 0 ∘ − 6 0 ∘ , so we can use the properties of cosine and sine to find the values of cos ( 30 0 ∘ ) and sin ( 30 0 ∘ ) .

cos ( 30 0 ∘ ) = cos ( 6 0 ∘ ) = 2 1 ​
sin ( 30 0 ∘ ) = − sin ( 6 0 ∘ ) = − 2 3 ​ ​

Calculate Components Now, substitute these values back into the formulas for a and b :

a = 24 × 2 1 ​ = 12
b = 24 × ( − 2 3 ​ ​ ) = − 12 3 ​

Final Vector Form Therefore, the vector v in the form ai + bj is:

v = 12 i − 12 3 ​ j
Examples
Understanding vectors is crucial in fields like physics and engineering. For instance, when analyzing projectile motion, we decompose the initial velocity vector into horizontal and vertical components to predict the range and maximum height of the projectile. Similarly, in navigation, vectors represent the magnitude and direction of movement, allowing for precise tracking and course correction.

Answered by GinnyAnswer | 2025-07-07

The vector v with a magnitude of 24 and making an angle of 300° with the positive x-axis can be expressed as 12 i − 12 3 ​ j .
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Answered by Anonymous | 2025-07-24