If \(\vec r\) = xî + yĵ + zk̂, then what is \(\vec r\) . (î + ĵ + k̂ ) equal to?
(x + y + z)
Understanding Vector Dot Product: A Step-by-Step Solution
The question asks us to find the dot product of a given vector \(\vec r\) and the vector \( (\hat i + \hat j + \hat k) \). The vector \(\vec r\) is defined as \( \vec r = x\hat i + y\hat j + z\hat k \).
Let's break down the process of calculating the dot product between two vectors in Cartesian coordinates.
What is a Dot Product?
The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. In Cartesian coordinates, the dot product is calculated by summing the products of the corresponding components of the two vectors.
Calculating the Dot Product
We are given the vector \(\vec r\):
\( \vec r = x\hat i + y\hat j + z\hat k \)
The second vector is:
\( \vec v = \hat i + \hat j + \hat k \)
Note that the second vector can also be written as:
\( \vec v = 1\hat i + 1\hat j + 1\hat k \)
The formula for the dot product of two vectors \(\vec a = a_x\hat i + a_y\hat j + a_z\hat k\) and \(\vec b = b_x\hat i + b_y\hat j + b_z\hat k\) is:
\( \vec a \cdot \vec b = a_x b_x + a_y b_y + a_z b_z \)
Applying this formula to our vectors \(\vec r\) and \( (\hat i + \hat j + \hat k) \):
- The x-component of \(\vec r\) is \(x\), and the x-component of \((\hat i + \hat j + \hat k)\) is \(1\). Their product is \(x \times 1 = x\).
- The y-component of \(\vec r\) is \(y\), and the y-component of \((\hat i + \hat j + \hat k)\) is \(1\). Their product is \(y \times 1 = y\).
- The z-component of \(\vec r\) is \(z\), and the z-component of \((\hat i + \hat j + \hat k)\) is \(1\). Their product is \(z \times 1 = z\).
Summing these products gives the dot product:
\( \vec r \cdot (\hat i + \hat j + \hat k) = (x)(1) + (y)(1) + (z)(1) \)
\( \vec r \cdot (\hat i + \hat j + \hat k) = x + y + z \)
Final Result of the Dot Product Calculation
The dot product \(\vec r\) . \( (\hat i + \hat j + \hat k) \) is equal to \( x + y + z \).
Let's compare this result with the given options:
| Option | Expression |
|---|---|
| 1 | \(x\) |
| 2 | \(x + y\) |
| 3 | \( -(x + y + z) \) |
| 4 | \( x + y + z \) |
Our calculated result, \( x + y + z \), matches Option 4.
Revision Table: Key Concepts in Vector Algebra
| Concept | Description |
|---|---|
| Vector | A quantity having magnitude and direction, represented geometrically by an arrow. |
| Unit Vectors (\(\hat i, \hat j, \hat k\)) | Vectors of magnitude 1 along the positive x, y, and z axes respectively. |
| Position Vector (\(\vec r\)) | A vector representing the position of a point in space relative to an origin, often written as \(x\hat i + y\hat j + z\hat k\). |
| Dot Product (Scalar Product) | An operation on two vectors that returns a scalar quantity. For \(\vec a = a_x\hat i + a_y\hat j + a_z\hat k\) and \(\vec b = b_x\hat i + b_y\hat j + b_z\hat k\), \(\vec a \cdot \vec b = a_x b_x + a_y b_y + a_z b_z\). |
| Properties of Dot Product | Commutative (\(\vec a \cdot \vec b = \vec b \cdot \vec a\)), Distributive (\(\vec a \cdot (\vec b + \vec c) = \vec a \cdot \vec b + \vec a \cdot \vec c\)), relates to the angle between vectors (\(\vec a \cdot \vec b = |\vec a| |\vec b| \cos \theta\)). |
Additional Information: Applications of the Dot Product
The dot product is a fundamental operation in physics and engineering. Some common applications include:
- Work Done: In physics, the work done by a force \(\vec F\) over a displacement \(\vec d\) is given by the dot product \(W = \vec F \cdot \vec d = |\vec F| |\vec d| \cos \theta\).
- Calculating the Angle Between Two Vectors: The dot product formula \(\vec a \cdot \vec b = |\vec a| |\vec b| \cos \theta\) can be rearranged to find the angle \(\theta\) between the vectors: \( \cos \theta = \frac{\vec a \cdot \vec b}{|\vec a| |\vec b|} \).
- Projection of One Vector onto Another: The scalar projection of vector \(\vec a\) onto vector \(\vec b\) is given by \( \frac{\vec a \cdot \vec b}{|\vec b|} \). This tells us how much of vector \(\vec a\) acts in the direction of vector \(\vec b\).
- Checking for Orthogonality: If the dot product of two non-zero vectors is zero (\(\vec a \cdot \vec b = 0\)), then the vectors are orthogonal (perpendicular) because \(\cos 90^\circ = 0\).
Understanding the dot product is crucial for solving problems involving vector projections, work, power, and angles in 2D and 3D space.
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