If \(|\vec a|\) = 3, \(\left| {\vec b} \right| = 4\) and \(\left| {\vec a - \vec b} \right| = 5,\) then what is the value of \(\left| {\vec a + \vec b} \right|\) ?
5
Understanding the Vector Problem
The question provides information about two vectors, \(\vec a\) and \(\vec b\). We are given the magnitudes of the individual vectors, \(|\vec a|\) and \(|\vec b|\), and the magnitude of their difference, \(|\vec a - \vec b|\). Our goal is to find the magnitude of their sum, \(|\vec a + \vec b|\).
Specifically, we have:
- Magnitude of vector \(\vec a\): \(|\vec a| = 3\)
- Magnitude of vector \(\vec b\): \(|\vec b| = 4\)
- Magnitude of the difference \(\vec a - \vec b\): \(|\vec a - \vec b| = 5\)
We need to calculate \(|\vec a + \vec b|\).
Applying Vector Magnitude Formulas
To solve this, we can use the formulas relating the magnitudes of vector sums and differences to the magnitudes of the individual vectors and the angle between them. Let \(\theta\) be the angle between vectors \(\vec a\) and \(\vec b\).
The formula for the magnitude squared of the difference of two vectors is:
\[|\vec a - \vec b|^2 = |\vec a|^2 + |\vec b|^2 - 2|\vec a||\vec b|\cos\theta\]
The formula for the magnitude squared of the sum of two vectors is:
\[|\vec a + \vec b|^2 = |\vec a|^2 + |\vec b|^2 + 2|\vec a||\vec b|\cos\theta\]
Finding the Relationship Between Vectors
We can use the given information about \(|\vec a - \vec b|\) to find the value of the term \(2|\vec a||\vec b|\cos\theta\).
Substitute the given values into the difference formula:
\[5^2 = 3^2 + 4^2 - 2|\vec a||\vec b|\cos\theta\]
Calculate the squares:
\[25 = 9 + 16 - 2|\vec a||\vec b|\cos\theta\]
Sum the numbers on the right side:
\[25 = 25 - 2|\vec a||\vec b|\cos\theta\]
Now, rearrange the equation to find \(2|\vec a||\vec b|\cos\theta\):
\[2|\vec a||\vec b|\cos\theta = 25 - 25\]
\[2|\vec a||\vec b|\cos\theta = 0\]
Since \(|\vec a| = 3\) and \(|\vec b| = 4\) are non-zero, this equation implies that \(\cos\theta\) must be 0. This means the angle \(\theta\) between vectors \(\vec a\) and \(\vec b\) is 90 degrees (or \(\pi/2\) radians). In other words, the vectors \(\vec a\) and \(\vec b\) are perpendicular.
Calculating the Magnitude of the Sum
Now that we know \(2|\vec a||\vec b|\cos\theta = 0\), we can substitute this into the formula for the magnitude squared of the sum:
\[|\vec a + \vec b|^2 = |\vec a|^2 + |\vec b|^2 + 2|\vec a||\vec b|\cos\theta\]
Substitute the values \(|\vec a| = 3\), \(|\vec b| = 4\), and \(2|\vec a||\vec b|\cos\theta = 0\):
\[|\vec a + \vec b|^2 = 3^2 + 4^2 + 0\]
\[|\vec a + \vec b|^2 = 9 + 16\]
\[|\vec a + \vec b|^2 = 25\]
To find the magnitude \(|\vec a + \vec b|\), take the square root of both sides. Since magnitude is always non-negative:
\[|\vec a + \vec b| = \sqrt{25}\]
\[|\vec a + \vec b| = 5\]
Alternative Perspective: Pythagorean Triplet
Notice the given magnitudes: \(|\vec a| = 3\), \(|\vec b| = 4\), and \(|\vec a - \vec b| = 5\). These numbers form a Pythagorean triplet (\(3^2 + 4^2 = 9 + 16 = 25 = 5^2\)).
The formula for the magnitude of the difference is \(|\vec a - \vec b|^2 = |\vec a|^2 + |\vec b|^2 - 2\vec a \cdot \vec b\), where \(\vec a \cdot \vec b = |\vec a||\vec b|\cos\theta\). If \(|\vec a - \vec b|^2 = |\vec a|^2 + |\vec b|^2\), this implies \(2\vec a \cdot \vec b = 0\), which means \(\vec a \cdot \vec b = 0\). A dot product of zero for non-zero vectors means the vectors are perpendicular (\(\theta = 90^\circ\)).
When vectors \(\vec a\) and \(\vec b\) are perpendicular, the formula for the magnitude of the sum simplifies:
\[|\vec a + \vec b|^2 = |\vec a|^2 + |\vec b|^2 + 2\vec a \cdot \vec b\]
Since \(\vec a \cdot \vec b = 0\) for perpendicular vectors:
\[|\vec a + \vec b|^2 = |\vec a|^2 + |\vec b|^2\]
This is the Pythagorean theorem! So, if the magnitudes \(|\vec a|\), \(|\vec b|\), and \(|\vec a - \vec b|\) form a Pythagorean triplet with \(|\vec a - \vec b|\) as the hypotenuse (which is the case here: \(3^2 + 4^2 = 5^2\)), the vectors are perpendicular. When vectors are perpendicular, the magnitude of their sum also follows the Pythagorean theorem:
\[|\vec a + \vec b| = \sqrt{|\vec a|^2 + |\vec b|^2}\]
Using the given values:
\[|\vec a + \vec b| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
Both methods lead to the same result.
Summary of the Solution
Given \(|\vec a| = 3\), \(|\vec b| = 4\), and \(|\vec a - \vec b| = 5\). We found that the vectors \(\vec a\) and \(\vec b\) are perpendicular because \(|\vec a|^2 + |\vec b|^2 = 3^2 + 4^2 = 9 + 16 = 25 = 5^2 = |\vec a - \vec b|^2\). When vectors are perpendicular, the magnitude of their sum is found using the Pythagorean theorem: \(|\vec a + \vec b| = \sqrt{|\vec a|^2 + |\vec b|^2}\). Thus, \(|\vec a + \vec b| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\).
| Given Information | Formula Used | Calculation | Result |
|---|---|---|---|
| \(|\vec a| = 3\) | \(|\vec a - \vec b|^2 = |\vec a|^2 + |\vec b|^2 - 2|\vec a||\vec b|\cos\theta\) | \(5^2 = 3^2 + 4^2 - 2|\vec a||\vec b|\cos\theta\) | \(2|\vec a||\vec b|\cos\theta = 0 \implies \cos\theta = 0\) |
| \(|\vec b| = 4\) | \(|\vec a + \vec b|^2 = |\vec a|^2 + |\vec b|^2 + 2|\vec a||\vec b|\cos\theta\) | \(|\vec a + \vec b|^2 = 3^2 + 4^2 + 0\) | \(|\vec a + \vec b| = 5\) |
| \(|\vec a - \vec b| = 5\) | Pythagorean Theorem for perpendicular vectors: \(|\vec a + \vec b| = \sqrt{|\vec a|^2 + |\vec b|^2}\) | \(|\vec a + \vec b| = \sqrt{3^2 + 4^2}\) | \(|\vec a + \vec b| = 5\) |
The final answer is 5.
Revision Table: Key Concepts for Vector Problems
| Concept | Description | Formula |
|---|---|---|
| Vector Magnitude | The length of a vector. Denoted by \(|\vec v|\). | If \(\vec v = (v_x, v_y)\), \(|\vec v| = \sqrt{v_x^2 + v_y^2}\). In 3D, \(|\vec v| = \sqrt{v_x^2 + v_y^2 + v_z^2}\). |
| Dot Product | A scalar value related to the angle between two vectors. \(\vec a \cdot \vec b = |\vec a||\vec b|\cos\theta\). | If \(\vec a = (a_x, a_y)\) and \(\vec b = (b_x, b_y)\), \(\vec a \cdot \vec b = a_xb_x + a_yb_y\). |
| Magnitude of Sum | The length of the resultant vector \(\vec a + \vec b\). | \(|\vec a + \vec b|^2 = |\vec a|^2 + |\vec b|^2 + 2\vec a \cdot \vec b = |\vec a|^2 + |\vec b|^2 + 2|\vec a||\vec b|\cos\theta\) |
| Magnitude of Difference | The length of the resultant vector \(\vec a - \vec b\). | \(|\vec a - \vec b|^2 = |\vec a|^2 + |\vec b|^2 - 2\vec a \cdot \vec b = |\vec a|^2 + |\vec b|^2 - 2|\vec a||\vec b|\cos\theta\) |
| Perpendicular Vectors | Two non-zero vectors \(\vec a\) and \(\vec b\) are perpendicular if the angle between them is 90°. | \(\vec a \cdot \vec b = 0\). This means \(|\vec a||\vec b|\cos\theta = 0\), and since magnitudes are non-zero, \(\cos\theta = 0\). |
Additional Information: Geometric Interpretation of Vector Addition and Subtraction
Vector addition (\(\vec a + \vec b\)) and subtraction (\(\vec a - \vec b\)) have geometric interpretations that can be helpful. Imagine vectors \(\vec a\) and \(\vec b\) originating from the same point.
- Vector Addition: The sum \(\vec a + \vec b\) is the diagonal of the parallelogram formed by \(\vec a\) and \(\vec b\), starting from the common origin. The magnitude \(|\vec a + \vec b|\) is the length of this diagonal.
- Vector Subtraction: The difference \(\vec a - \vec b\) is the other diagonal of the same parallelogram. It starts from the endpoint of \(\vec b\) and ends at the endpoint of \(\vec a\). The magnitude \(|\vec a - \vec b|\) is the length of this diagonal. Note that \(\vec a - \vec b = \vec a + (-\vec b)\), where \(-\vec b\) is a vector with the same magnitude as \(\vec b\) but opposite direction.
In our specific problem, \(|\vec a|=3\), \(|\vec b|=4\), and \(|\vec a - \vec b|=5\). Since \(3^2 + 4^2 = 5^2\), the triangle formed by vectors \(\vec a\), \(\vec b\), and \(\vec a - \vec b\) (when arranged head-to-tail) is a right-angled triangle. This confirms that the angle between \(\vec a\) and \(\vec b\) is 90 degrees.
When \(\vec a\) and \(\vec b\) are perpendicular, the parallelogram they form is a rectangle. The diagonals of a rectangle have equal length. One diagonal represents \(\vec a + \vec b\) (or \(\vec b + \vec a\)), and the other represents \(\vec a - \vec b\) (or \(\vec b - \vec a\)). Therefore, if the vectors are perpendicular, \(|\vec a + \vec b| = |\vec a - \vec b|\). Since \(|\vec a - \vec b| = 5\), it directly follows that \(|\vec a + \vec b| = 5\).
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