Questions: Vektor A dan B berada di bidang xy. Vektor A mempunyai besaran 8 satuan dan sudut 130°. Vektor B memiliki komponen Bx=-7,72 dan By=-9,20. Berapa sudut antara sumbu y negative dengan produk vektor dari A x B ? A. 30° B. 45° C. 70° D. 90°

Vektor A dan B berada di bidang xy. Vektor A mempunyai besaran 8 satuan dan sudut 130°. Vektor B memiliki komponen Bx=-7,72 dan By=-9,20. Berapa sudut antara sumbu y negative dengan produk vektor dari A x B ? A. 30° B. 45° C. 70° D. 90°
Transcript text: Vektor $\vec{A}$ dan $\vec{B}$ berada di bidang xy. Vektor $\vec{A}$ mempunyai besaran 8 satuan dan sudut $130^{\circ}$. Vektor $\vec{B}$ memiliki komponen $\mathrm{B}_{\mathrm{x}}=-7,72$ dan $\mathrm{B}_{\mathrm{y}}=-9,20$. Berapa sudut antara sumbu y negative dengan produk vektor dari $\vec{A} \times \vec{B}$ ? A. $30^{\circ}$ B. $45^{\circ}$ C. $70^{\circ}$ D. $90^{\circ}$
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Solution

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Solution Steps

Step 1: Determine the Components of Vector \(\vec{A}\)

Given:

  • Magnitude of \(\vec{A}\) is 8 units.
  • Angle of \(\vec{A}\) is \(130^\circ\).

We can find the components of \(\vec{A}\) using trigonometric functions: \[ A_x = 8 \cos(130^\circ) \] \[ A_y = 8 \sin(130^\circ) \]

Using the values of \(\cos(130^\circ) \approx -0.6428\) and \(\sin(130^\circ) \approx 0.7660\): \[ A_x = 8 \times (-0.6428) = -5.1424 \] \[ A_y = 8 \times 0.7660 = 6.1280 \]

Step 2: Determine the Components of Vector \(\vec{B}\)

Given:

  • \(B_x = -7.72\)
  • \(B_y = -9.20\)
Step 3: Calculate the Cross Product \(\vec{A} \times \vec{B}\)

The cross product of two vectors \(\vec{A}\) and \(\vec{B}\) in the xy-plane is given by: \[ \vec{A} \times \vec{B} = (A_x B_y - A_y B_x) \hat{z} \]

Substituting the values: \[ \vec{A} \times \vec{B} = (-5.1424 \times -9.20) - (6.1280 \times -7.72) \] \[ = 47.3096 + 47.1104 \] \[ = 94.4200 \hat{z} \]

Step 4: Determine the Angle with the Negative y-Axis

The cross product \(\vec{A} \times \vec{B}\) points in the positive z-direction. The negative y-axis lies in the xy-plane, and the z-axis is perpendicular to this plane.

The angle between the positive z-axis and the negative y-axis is \(90^\circ\).

Final Answer

\[ \boxed{90^\circ} \]

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