a vector A is along the positive X-axis. if B is another vector such...
**Introduction:**
In this scenario, we are given that vector A is along the positive X-axis and we need to find another vector B such that the dot product of A and B is zero. The dot product of two vectors is calculated by multiplying their corresponding components and adding them together. If the dot product of two vectors is zero, it means that the vectors are orthogonal or perpendicular to each other.
**Explanation:**
To find a vector B that satisfies the given condition, we can consider the properties of the dot product and the fact that vector A is along the positive X-axis.
**1. Understanding the dot product:**
The dot product of two vectors A and B is given by the formula:
A • B = |A| |B| cos θ
where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them.
**2. Vector A along the positive X-axis:**
Since vector A is along the positive X-axis, its components in the Y and Z directions are zero. Let's assume the components of vector B as (Bx, By, Bz).
**3. Calculation of dot product:**
The dot product of vector A and B can be calculated as follows:
A • B = Ax*Bx + Ay*By + Az*Bz
Since Ay and Az are zero, the equation simplifies to:
A • B = Ax*Bx
**4. Condition for A • B = 0:**
To satisfy the given condition of A • B = 0, we need to find a value for Bx such that the equation Ax*Bx = 0 holds true. This implies that either Ax or Bx must be zero.
**5. Possible values of B:**
Based on the above condition, there are two possible scenarios:
a) If Ax = 0, then Bx can be any arbitrary value since any value multiplied by zero will result in zero.
b) If Bx = 0, then Ax can be any non-zero value, as long as Ay and Az are also non-zero to ensure vector B is not parallel to the X-axis.
**Conclusion:**
In conclusion, if vector A is along the positive X-axis, then vector B can be chosen in two ways: either Ax = 0 and Bx can be any value, or Bx = 0 and Ax can be any non-zero value with non-zero components in the Y and Z directions. These scenarios satisfy the condition of A • B = 0, indicating that vectors A and B are perpendicular to each other.
a vector A is along the positive X-axis. if B is another vector such...
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