Scalars and Vectors - AP Physics C: Mechanics
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What is the effect of multiplying a vector by a scalar?
What is the effect of multiplying a vector by a scalar?
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It changes the magnitude, not the direction. Scalar multiplication scales size but preserves direction.
It changes the magnitude, not the direction. Scalar multiplication scales size but preserves direction.
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Calculate the dot product of $\textbf{A} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\textbf{B} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.
Calculate the dot product of $\textbf{A} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\textbf{B} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.
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The dot product is 11. Calculate $1 \times 3 + 2 \times 4 = 3 + 8 = 11$.
The dot product is 11. Calculate $1 \times 3 + 2 \times 4 = 3 + 8 = 11$.
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Determine if acceleration is a scalar or vector quantity.
Determine if acceleration is a scalar or vector quantity.
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Acceleration is a vector quantity. Acceleration has both magnitude and direction.
Acceleration is a vector quantity. Acceleration has both magnitude and direction.
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State the formula for the dot product of vectors $\textbf{A}$ and $\textbf{B}$.
State the formula for the dot product of vectors $\textbf{A}$ and $\textbf{B}$.
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$\textbf{A} \bullet \textbf{B} = A_xB_x + A_yB_y$. Multiply corresponding components and sum them.
$\textbf{A} \bullet \textbf{B} = A_xB_x + A_yB_y$. Multiply corresponding components and sum them.
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What does it mean if two vectors are orthogonal?
What does it mean if two vectors are orthogonal?
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Their dot product is zero. Orthogonal means perpendicular, so no projection exists.
Their dot product is zero. Orthogonal means perpendicular, so no projection exists.
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Calculate the dot product of $\textbf{A} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\textbf{B} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.
Calculate the dot product of $\textbf{A} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\textbf{B} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.
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The dot product is 11. Calculate $1 \times 3 + 2 \times 4 = 3 + 8 = 11$.
The dot product is 11. Calculate $1 \times 3 + 2 \times 4 = 3 + 8 = 11$.
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What is the magnitude of $\textbf{A} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$?
What is the magnitude of $\textbf{A} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$?
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The magnitude is 0. Zero vector has magnitude $\sqrt{0^2 + 0^2} = 0$.
The magnitude is 0. Zero vector has magnitude $\sqrt{0^2 + 0^2} = 0$.
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Identify the vector form of Newton's second law.
Identify the vector form of Newton's second law.
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$\textbf{F} = m\textbf{a}$. Force vector equals mass times acceleration vector.
$\textbf{F} = m\textbf{a}$. Force vector equals mass times acceleration vector.
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What is the cross product of parallel vectors?
What is the cross product of parallel vectors?
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The cross product is zero. Parallel vectors have no perpendicular component.
The cross product is zero. Parallel vectors have no perpendicular component.
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What is the physical significance of the cross product?
What is the physical significance of the cross product?
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It gives a vector perpendicular to both original vectors. Creates a vector orthogonal to both input vectors.
It gives a vector perpendicular to both original vectors. Creates a vector orthogonal to both input vectors.
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Determine if force is a scalar or vector quantity.
Determine if force is a scalar or vector quantity.
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Force is a vector quantity. Force has both magnitude and direction.
Force is a vector quantity. Force has both magnitude and direction.
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Convert the vector $\textbf{v} = \begin{bmatrix} 6 \\ 8 \end{bmatrix}$ to a unit vector.
Convert the vector $\textbf{v} = \begin{bmatrix} 6 \\ 8 \end{bmatrix}$ to a unit vector.
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Unit vector is $\begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}$. Magnitude is $\sqrt{6^2 + 8^2} = 10$, so divide by 10.
Unit vector is $\begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}$. Magnitude is $\sqrt{6^2 + 8^2} = 10$, so divide by 10.
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Find the magnitude of the resultant vector $\textbf{R} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.
Find the magnitude of the resultant vector $\textbf{R} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.
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The magnitude is 5. Use $ |\textbf{R}| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 $.
The magnitude is 5. Use $ |\textbf{R}| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 $.
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Convert the vector $\textbf{v} = \begin{bmatrix} 6 \\ 8 \end{bmatrix}$ to a unit vector.
Convert the vector $\textbf{v} = \begin{bmatrix} 6 \\ 8 \end{bmatrix}$ to a unit vector.
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Unit vector is $\begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}$. Magnitude is $\sqrt{6^2 + 8^2} = 10$, so divide by 10.
Unit vector is $\begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}$. Magnitude is $\sqrt{6^2 + 8^2} = 10$, so divide by 10.
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Express the vector $\textbf{v} = \begin{bmatrix} -4 \\ 3 \end{bmatrix}$ in terms of magnitude and direction.
Express the vector $\textbf{v} = \begin{bmatrix} -4 \\ 3 \end{bmatrix}$ in terms of magnitude and direction.
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Magnitude is 5; direction is $\tan^{-1}(-\frac{3}{4})$. Magnitude is $\sqrt{(-4)^2 + 3^2} = 5$; angle from positive x-axis.
Magnitude is 5; direction is $\tan^{-1}(-\frac{3}{4})$. Magnitude is $\sqrt{(-4)^2 + 3^2} = 5$; angle from positive x-axis.
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What is a vector quantity?
What is a vector quantity?
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A vector is a quantity with both magnitude and direction. Direction distinguishes vectors from scalars.
A vector is a quantity with both magnitude and direction. Direction distinguishes vectors from scalars.
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What is a scalar quantity?
What is a scalar quantity?
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A scalar is a quantity with only magnitude. It has no direction component, only size.
A scalar is a quantity with only magnitude. It has no direction component, only size.
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Calculate the magnitude of $\textbf{v} = \begin{bmatrix} 7 \\ 24 \end{bmatrix}$
Calculate the magnitude of $\textbf{v} = \begin{bmatrix} 7 \\ 24 \end{bmatrix}$
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The magnitude is 25. Use $|\textbf{v}| = \sqrt{7^2 + 24^2} = \sqrt{625} = 25$
The magnitude is 25. Use $|\textbf{v}| = \sqrt{7^2 + 24^2} = \sqrt{625} = 25$
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What is the magnitude of $\textbf{A} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$?
What is the magnitude of $\textbf{A} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$?
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The magnitude is 0. Zero vector has magnitude $\sqrt{0^2 + 0^2} = 0$.
The magnitude is 0. Zero vector has magnitude $\sqrt{0^2 + 0^2} = 0$.
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Identify the vector form of Newton's second law.
Identify the vector form of Newton's second law.
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$\textbf{F} = m\textbf{a}$. Force vector equals mass times acceleration vector.
$\textbf{F} = m\textbf{a}$. Force vector equals mass times acceleration vector.
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What does it mean if two vectors are orthogonal?
What does it mean if two vectors are orthogonal?
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Their dot product is zero. Orthogonal means perpendicular, so no projection exists.
Their dot product is zero. Orthogonal means perpendicular, so no projection exists.
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What is the cross product of parallel vectors?
What is the cross product of parallel vectors?
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The cross product is zero. Parallel vectors have no perpendicular component.
The cross product is zero. Parallel vectors have no perpendicular component.
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What is the zero vector?
What is the zero vector?
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A vector with zero magnitude and no specific direction. It's the additive identity for vector operations.
A vector with zero magnitude and no specific direction. It's the additive identity for vector operations.
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Find the angle between $\textbf{A} = \begin{bmatrix} 1 \\ 0 \\ \textend{bmatrix}$ and $\textbf{B} = \begin{bmatrix} 0 \\ 1 \\ \textend{bmatrix}$.
Find the angle between $\textbf{A} = \begin{bmatrix} 1 \\ 0 \\ \textend{bmatrix}$ and $\textbf{B} = \begin{bmatrix} 0 \\ 1 \\ \textend{bmatrix}$.
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The angle is 90 degrees. Perpendicular unit vectors have dot product zero.
The angle is 90 degrees. Perpendicular unit vectors have dot product zero.
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State the component form of a vector given magnitude and direction.
State the component form of a vector given magnitude and direction.
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$\textbf{v} = v(\text{cos} \theta \textbf{i} + \text{sin} \theta \textbf{j})$. Uses trigonometric components with magnitude $v$ and angle $\theta$.
$\textbf{v} = v(\text{cos} \theta \textbf{i} + \text{sin} \theta \textbf{j})$. Uses trigonometric components with magnitude $v$ and angle $\theta$.
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How is the direction of a vector represented?
How is the direction of a vector represented?
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By the angle it makes with a reference axis. Direction angle is measured from positive x-axis.
By the angle it makes with a reference axis. Direction angle is measured from positive x-axis.
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Identify the scalar quantity: speed, velocity, force.
Identify the scalar quantity: speed, velocity, force.
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Speed is a scalar quantity. Speed lacks direction, unlike velocity and force.
Speed is a scalar quantity. Speed lacks direction, unlike velocity and force.
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What is the physical significance of the cross product?
What is the physical significance of the cross product?
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It gives a vector perpendicular to both original vectors. Creates a vector orthogonal to both input vectors.
It gives a vector perpendicular to both original vectors. Creates a vector orthogonal to both input vectors.
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Identify the result of multiplying a vector by -1.
Identify the result of multiplying a vector by -1.
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It reverses the direction of the vector. Negative scalar multiplication flips the vector direction.
It reverses the direction of the vector. Negative scalar multiplication flips the vector direction.
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What is meant by the magnitude of a vector?
What is meant by the magnitude of a vector?
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Magnitude is the length or size of the vector. Magnitude is the numerical value of vector size.
Magnitude is the length or size of the vector. Magnitude is the numerical value of vector size.
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