Magnitude of vector combination
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Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes 2 and 1 respectively. If the angle between $\vec{A}$ and $\vec{B}$ is $60^\circ$, then which of the following vectors may be equal to $\frac{\vec{A}}{2} - \vec{B}$.
This question includes visual content: The image shows two vectors, A and B. Vector A points diagonally upwards to the right, and vector B points horizontally to the right. The angle between them is stated as 60 degrees. Below the text, there are four options labeled (A), (B), (C), and (D), each displaying an arrow pointing in different directions.
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Step by Step Written Solution
Hi Gnyanesh, let's solve this vector subtraction problem together.
Vector Subtraction
Let's first define a coordinate system. We can align vector B along the positive x-axis. Since its magnitude is one, we can easily write it in unit vector notation.
Vector Representation
Thus, vector B is simply equal to one times i-cap.
Vector A has a magnitude of two and makes an angle of sixty degrees with vector B, which is along the positive x-axis.
Substituting the magnitude of two and the angle of sixty degrees, we get two cos sixty i-cap plus two sin sixty j-cap.
Since cosine of sixty degrees is half, and sine of sixty degrees is root three over two, this simplifies to one i-cap plus root three j-cap.
Now, let's divide vector A by two. This gives us half of i-cap plus root three over two of j-cap.
Let's visually construct the vector C, which is equal to vector A over two minus vector B. We draw vector A over two, and then add negative vector B, which points to the left.
Geometric Addition
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