Given the vector A = 4.00i + 7.00j, find the magnitude of the vector. Given the vector B = 5.00i - 2.00j, find the magnitude of the vector. Write an expression for the vector diffe... Given the vector A = 4.00i + 7.00j, find the magnitude of the vector. Given the vector B = 5.00i - 2.00j, find the magnitude of the vector. Write an expression for the vector difference A - B using unit vectors.
Understand the Problem
The question is asking to find the magnitudes of two given vectors in parts A and B, and then to express the difference of those two vectors in terms of unit vectors in part C. The magnitude of a vector can be calculated using the formula √(x² + y²), where x and y are the components of the vector. In part C, the difference between the vectors A and B is to be determined.
Answer
Magnitude of $\vec{A} = \sqrt{65}$, Magnitude of $\vec{B} = \sqrt{29}$, $\vec{A} - \vec{B} = -1.00 \hat{i} + 9.00 \hat{j}$
Answer for screen readers
The magnitudes are:
Magnitude of $\vec{A}$: $|\vec{A}| = \sqrt{65}$
Magnitude of $\vec{B}$: $|\vec{B}| = \sqrt{29}$
Difference $\vec{A} - \vec{B}$: $\vec{A} - \vec{B} = -1.00 \hat{i} + 9.00 \hat{j}$
Steps to Solve
- Find the Magnitude of Vector A
To find the magnitude of vector $\vec{A} = 4.00 \hat{i} + 7.00 \hat{j}$, use the formula for the magnitude of a vector:
$$ |\vec{A}| = \sqrt{x^2 + y^2} $$
Here, $x = 4.00$ and $y = 7.00$.
Calculate:
$$ |\vec{A}| = \sqrt{(4.00)^2 + (7.00)^2} = \sqrt{16 + 49} = \sqrt{65} $$
- Find the Magnitude of Vector B
Next, find the magnitude of vector $\vec{B} = 5.00 \hat{i} - 2.00 \hat{j}$. Using the same formula:
$$ |\vec{B}| = \sqrt{x^2 + y^2} $$
Here, $x = 5.00$ and $y = -2.00$.
Calculate:
$$ |\vec{B}| = \sqrt{(5.00)^2 + (-2.00)^2} = \sqrt{25 + 4} = \sqrt{29} $$
- Find the Difference of Vectors A and B
To find the difference $\vec{A} - \vec{B}$:
$$ \vec{A} - \vec{B} = (4.00 \hat{i} + 7.00 \hat{j}) - (5.00 \hat{i} - 2.00 \hat{j}) $$
Distributing the negative sign:
$$ \vec{A} - \vec{B} = (4.00 - 5.00) \hat{i} + (7.00 + 2.00) \hat{j} $$
This simplifies to:
$$ \vec{A} - \vec{B} = -1.00 \hat{i} + 9.00 \hat{j} $$
The magnitudes are:
Magnitude of $\vec{A}$: $|\vec{A}| = \sqrt{65}$
Magnitude of $\vec{B}$: $|\vec{B}| = \sqrt{29}$
Difference $\vec{A} - \vec{B}$: $\vec{A} - \vec{B} = -1.00 \hat{i} + 9.00 \hat{j}$
More Information
The magnitudes of vectors are crucial in physics and engineering to determine the length of the vectors in space. Vector subtraction helps in finding relative positions or directions.
Tips
- Forgetting to square the components before summing for magnitude.
- Misapplying the negative sign during vector subtraction.
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