AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Vector Scalar Multiplication Problem: Find Magnitude and Direction of Scaled Vectors
Physics
High School Grade 11
Question Content
A vector $\vec{d}$ has a magnitude 1.9 m and is directed south. What are (a) the magnitude and (b) the direction of the vector $9.7\vec{d}$? What are (c) the magnitude and (d) the direction of the vector $-9.9\vec{d}$?
Correct Answer
(a) 18.43 m; (b) South; (c) 18.81 m; (d) North
Detailed Solution Steps
1
Step 1: Solve for part (a): When a vector is multiplied by a positive scalar, the magnitude scales by the absolute value of the scalar, and direction stays the same. Calculate the magnitude: $|9.7\vec{d}| = 9.7 \\times |\vec{d}| = 9.7 \\times 1.9\\text{ m} = 18.43\\text{ m}$
2
Step 2: Solve for part (b): Since the scalar 9.7 is positive, the direction of $9.7\vec{d}$ is the same as $\vec{d}$, which is south.
3
Step 3: Solve for part (c): When a vector is multiplied by a negative scalar, the magnitude still scales by the absolute value of the scalar. Calculate the magnitude: $|-9.9\vec{d}| = |-9.9| \\times |\vec{d}| = 9.9 \\times 1.9\\text{ m} = 18.81\\text{ m}$
4
Step 4: Solve for part (d): The negative scalar $-9.9$ reverses the direction of the original vector. Since $\vec{d}$ points south, $-9.9\vec{d}$ points north.
Knowledge Points Involved
1
Scalar Multiplication of Vectors
When a vector is multiplied by a scalar, the magnitude of the resulting vector is the product of the absolute value of the scalar and the magnitude of the original vector. The direction remains the same if the scalar is positive, and reverses if the scalar is negative. This applies to all vector quantities (displacement, velocity, force, etc.) in physics.
2
Vector Magnitude
The magnitude of a vector is a non-negative scalar that represents the size or length of the vector, independent of its direction. For a scaled vector $k\vec{v}$, $|k\vec{v}| = |k| \\times |\vec{v}|$, where $|k|$ is the absolute value of the scalar and $|\vec{v}|$ is the magnitude of the original vector.
3
Vector Direction
Vector direction describes the orientation of the vector in space. Multiplying a vector by a positive scalar preserves its direction, while multiplying by a negative scalar inverts its direction (180 degrees opposite to the original orientation).
Loading solution...
