Vector A̅ = ŷ.3 + ẑ.2 and B̅ = x̂.5 + ŷ.8 extend from the origin. Find A̅.B̅ Choose the correct answer.
24
Understanding the Vector Dot Product
The question asks us to find the dot product of two vectors, $\vec{A}$ and $\vec{B}$, given in component form. The dot product, also known as the scalar product, is a fundamental operation in vector algebra that results in a scalar quantity.
The given vectors are:
- Vector $\vec{A} = \hat{y} \cdot 3 + \hat{z} \cdot 2$. In standard component notation ($\hat{i}, \hat{j}, \hat{k}$ for x, y, z axes respectively), this vector has no x-component, a y-component of 3, and a z-component of 2. So, $\vec{A} = 0\hat{i} + 3\hat{j} + 2\hat{k}$.
- Vector $\vec{B} = \hat{x} \cdot 5 + \hat{y} \cdot 8$. In standard component notation, this vector has an x-component of 5, a y-component of 8, and no z-component. So, $\vec{B} = 5\hat{i} + 8\hat{j} + 0\hat{k}$.
Calculating the Dot Product of Vectors
To find the dot product of two vectors $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ and $\vec{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}$, we use the formula:
$\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$
Let's apply this formula to the given vectors $\vec{A} = 0\hat{i} + 3\hat{j} + 2\hat{k}$ and $\vec{B} = 5\hat{i} + 8\hat{j} + 0\hat{k}$.
- $A_x = 0$, $A_y = 3$, $A_z = 2$
- $B_x = 5$, $B_y = 8$, $B_z = 0$
Now, substitute these values into the dot product formula:
$\vec{A} \cdot \vec{B} = (0)(5) + (3)(8) + (2)(0)$
Perform the multiplications:
$\vec{A} \cdot \vec{B} = 0 + 24 + 0$
Finally, sum the results:
$\vec{A} \cdot \vec{B} = 24$
The dot product of vectors $\vec{A}$ and $\vec{B}$ is 24.
Result of the Vector Dot Product Calculation
The calculated dot product $\vec{A} \cdot \vec{B}$ is 24. This is a scalar value, as expected for a dot product.
Let's compare this result with the given options:
- Option 1: 15
- Option 2: 16
- Option 3: 24
- Option 4: 6
Our calculated value, 24, matches Option 3.
Revision Table: Key Concepts for Vector Dot Product
| Concept | Description | Formula/Property |
|---|---|---|
| Vector | A quantity having both magnitude and direction. Represented by components ($\hat{i}, \hat{j}, \hat{k}$) or magnitude and angle. | $\vec{V} = V_x\hat{i} + V_y\hat{j} + V_z\hat{k}$ |
| Dot Product (Scalar Product) | An operation between two vectors that yields a scalar quantity. It measures how much one vector extends in the direction of the other. | $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)$ (where $\theta$ is angle between $\vec{A}$ and $\vec{B}$) $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ (using components) |
| Orthogonal Vectors | Two vectors are orthogonal (perpendicular) if their dot product is zero ($\vec{A} \cdot \vec{B} = 0$). | $\hat{i} \cdot \hat{j} = 0$, $\hat{j} \cdot \hat{k} = 0$, $\hat{k} \cdot \hat{i} = 0$ |
| Parallel Vectors | Two vectors are parallel if the magnitude of their dot product equals the product of their magnitudes ($|\vec{A} \cdot \vec{B}| = |\vec{A}| |\vec{B}|$). | $\hat{i} \cdot \hat{i} = 1$, $\hat{j} \cdot \hat{j} = 1$, $\hat{k} \cdot \hat{k} = 1$ |
Additional Information on Vector Dot Product Uses
The dot product has many important applications in physics and engineering:
- Work Done: In physics, the work done by a constant force $\vec{F}$ on an object moving through a displacement $\vec{d}$ is given by the dot product: $W = \vec{F} \cdot \vec{d}$.
- Finding the Angle Between 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 two 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 you the length of the component of $\vec{A}$ that lies along the direction of $\vec{B}$.
- Checking for Perpendicularity: As mentioned, if the dot product of two non-zero vectors is zero, they are perpendicular.
These applications highlight the significance of the dot product beyond just a mathematical operation; it provides meaningful physical and geometric information.
Important Questions from Vector Algebra
The value of the cross product \(\left( {\overrightarrow a - \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow b } \right)\) of two vectors \(\overrightarrow a - \overrightarrow b\) and \(\overrightarrow a + \overrightarrow b \) is:
If non - zero a, b, c are such that a + b + c = 0, then the value of \(\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab}\) is
If â and b̂ are unit vectors such that â + 2b̂ and 5â - 4b̂ are perpendicular to each other, then the angle between â and b̂ is
Vector a = 3i + 2j – 6k, vector b = 4i – 3j + k, angle between above vectors is
Two forces F1 and F2 are used to pull a car, which met an accident. The angle between the two force is θ. Find the value of θ for the resultant force is equal to \(\sqrt{(F_1^2 + F_2^2)}\)
