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Physics Chapter 2 : Topic 2
Dot and Cross Products of Vectors
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The dot product of two vectors produces:
Reasoning: The dot product gives a scalar value calculated as $\overrightarrow{\mathbf{A}}\mathbf{\cdot}\overrightarrow{\mathbf{B}}\mathbf{= AB\ cos\ \theta}$.
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The formula for dot product of two vectors is:
Reasoning: Dot product depends on the cosine of the angle between vectors.
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The dot product of two perpendicular vectors is:
Reasoning: For $\mathbf{\theta}\mathbf{=}\mathbf{90}^{\mathbf{o}}$,$\mathbf{\ }\mathbf{cos}\mathbf{90}^{\mathbf{o}}\mathbf{= 0}$.
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The dot product of two parallel vectors in the same direction is:
Reasoning: $\mathbf{cos\ }\mathbf{0}^{\mathbf{o}}\mathbf{= 1\ }$, so $\mathbf{A \cdot B = AB}$.
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If two vectors are in opposite directions, their dot product is:
Reasoning: For $\mathbf{180}^{\mathbf{o}}$,$\mathbf{\ }\mathbf{cos\ }\mathbf{180}^{\mathbf{o}}\mathbf{= - \ 1}$.
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Which physical quantity is calculated using dot product?
Reasoning: Work is given by $\mathbf{W =}\overrightarrow{\mathbf{F}}\mathbf{\cdot}\overrightarrow{\mathbf{s}}\mathbf{= Fs\ cos\ \theta}$.
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The dot product is also known as:
Reasoning: Dot product results in a scalar.
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The value of $\overrightarrow{\mathbf{A}}\mathbf{\cdot}\overrightarrow{\mathbf{B}}$ is maximum when the angle between them is:
Reasoning: Maximum value occurs when cosine is maximum.
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If $\mathbf{A = 5\ }$,$\mathbf{\ }\mathbf{B = 4}$, and angle between them is $\mathbf{60}^{\mathbf{o}}$, the dot product is:
Reasoning: $\mathbf{A \cdot B = AB\ cos\ }\mathbf{60}^{\mathbf{o}}\mathbf{= 5(4)(0.5) = 10}$
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The cross product of two vectors produces:
Reasoning: Cross product gives a vector perpendicular to both original vectors.
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The magnitude of cross product is:
Reasoning: Cross product depends on the sine of the angle.
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The direction of $\mathbf{(AB)}$ is determined by:
Reasoning: Right-hand rule gives the direction of cross product.
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The cross product of parallel vectors is:
Reasoning: $\mathbf{sin\ 0}^{\mathbf{o}}\mathbf{= 0}$
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The cross product of perpendicular vectors is:
Reasoning: $\mathbf{sin\ }\mathbf{90}^{\mathbf{0}}\mathbf{= 1}$, giving maximum value.
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Which physical quantity uses cross product?
Reasoning: Torque is given by $\overrightarrow{\mathbf{\tau}}\mathbf{=}\overrightarrow{\mathbf{r}}\mathbf{\times}\overrightarrow{\mathbf{F}}$.
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The cross product is also called:
Reasoning: Cross product produces a vector.
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If $\overrightarrow{\mathbf{A}}\mathbf{\times}\overrightarrow{\mathbf{B}}\mathbf{= 0}$, then vectors are:
Reasoning: Cross product is zero when angle is $\mathbf{0}^{\mathbf{o}}\mathbf{\ \ }$or $\mathbf{180}^{\mathbf{o}}$.
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If two vectors have magnitudes $\mathbf{6}$ and $\mathbf{8}$ and are perpendicular, their cross product magnitude is:
Reasoning: $\mathbf{AB\ sin\ }\mathbf{90}^{\mathbf{o}}\mathbf{= 6(8) = 48}$
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Which statement about dot product is correct?
Reasoning: The sign depends on $\mathbf{cos\ \theta}$.
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Which statement about cross product is correct?
Reasoning: Cross product depends on $\mathbf{sin\ \theta}$.
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The relation between dot product and angle is:
Reasoning: Dot product measures the projection of one vector on another.
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The relation between cross product and angle is:
Reasoning: Cross product depends on perpendicular components.
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Which operation is commutative?
Reasoning: $\mathbf{A\ \ .\ \ B = B\ \ .\ \ A}$
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Which operation is not commutative?
Reasoning: $\mathbf{A \times B = - \ (B \times A)}$
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The property of cross product is:
Reasoning: Reversing order reverses direction.
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The dot product of a vector with itself gives:
Reasoning: $\mathbf{A \cdot A =}\mathbf{A}^{\mathbf{2}}\mathbf{\ cos\ }\mathbf{0}^{\mathbf{o}}\mathbf{=}\mathbf{A}^{\mathbf{2}}\mathbf{\times 1 =}\mathbf{A}^{\mathbf{2}}$
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The cross product of a vector with itself is:
Reasoning: $\mathbf{A \times A =}\mathbf{A}^{\mathbf{2}}\mathbf{\ sin\ }\mathbf{0}^{\mathbf{o}}\mathbf{=}\mathbf{A}^{\mathbf{2}}\mathbf{\times 0}\mathbf{= 0}$
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A vector product result is always:
Reasoning: Cross product gives a vector normal to the plane containing the two vectors.
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If the angle between two vectors changes from $\mathbf{30}^{\mathbf{o}}\mathbf{\ }$to $\mathbf{90}^{\mathbf{o}}$, their cross product:
Reasoning: $\mathbf{sin\ }\mathbf{90}^{\mathbf{o}}\mathbf{= 1}$, maximum cross product.
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If the angle between two vectors changes from $\mathbf{0}^{\mathbf{o}}$ to $\mathbf{90}^{\mathbf{o}}$, their dot product:
Reasoning: Dot product changes from $\mathbf{(AB)}$ to zero.
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The work done by a force is zero when force and displacement are:
Reasoning: $\mathbf{W = Fs\ cos\ }\mathbf{90}^{\mathbf{o}}\mathbf{= 0}$
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Torque is maximum when force and position vectors are:
Reasoning: $\mathbf{\tau = rF\ sin\ }\mathbf{90}^{\mathbf{o}}$
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Which operation can represent projection of one vector on another?
Reasoning: Dot product measures the component of one vector along another.
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If two vectors have a negative dot product, the angle between them is:
Reasoning: Negative cosine occurs for angles greater than $\mathbf{90}^{\mathbf{o}}$.
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If two vectors have zero dot product, they are:
Reasoning: $\mathbf{A \cdot B = AB\ cos\ }\mathbf{90}^{\mathbf{o}}\mathbf{= 0}$
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The magnitude of cross product represents:
Reasoning: $\mathbf{AB\ sin\ \theta}$ gives parallelogram area.
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Which quantity is NOT obtained using vector products?
Reasoning: Work uses dot product; others commonly involve cross products.
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Angular momentum is expressed as:
Reasoning: Angular momentum is a cross product of position and momentum.
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If the angle between vectors is $\mathbf{90}^{\mathbf{o}}$, then:
Reasoning: Cross product reaches maximum value at $\mathbf{90}^{\mathbf{o}}$.
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The main difference between dot and cross products is:
Reasoning: Dot product produces scalar; cross product produces vector.
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