Solving Problems with Vectors and Velocity - Pre-Calculus
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What is the component form of the vector from $A(x_1,y_1)$ to $B(x_2,y_2)$?
What is the component form of the vector from $A(x_1,y_1)$ to $B(x_2,y_2)$?
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$\langle x_2-x_1,,y_2-y_1\rangle$. Subtract initial coordinates from terminal coordinates to get components.
$\langle x_2-x_1,,y_2-y_1\rangle$. Subtract initial coordinates from terminal coordinates to get components.
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Identify whether $\langle 1,2\rangle$ and $\langle 4,-2\rangle$ are perpendicular.
Identify whether $\langle 1,2\rangle$ and $\langle 4,-2\rangle$ are perpendicular.
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$\text{Yes, since }\langle 1,2\rangle\cdot\langle 4,-2\rangle=0$. $(1)(4) + (2)(-2) = 4 - 4 = 0$, so they are perpendicular.
$\text{Yes, since }\langle 1,2\rangle\cdot\langle 4,-2\rangle=0$. $(1)(4) + (2)(-2) = 4 - 4 = 0$, so they are perpendicular.
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What is the magnitude of a $2$D vector $\vec{v}=\langle a,b\rangle$?
What is the magnitude of a $2$D vector $\vec{v}=\langle a,b\rangle$?
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$|\vec{v}|=\sqrt{a^2+b^2}$. Apply the Pythagorean theorem to find the length of the vector.
$|\vec{v}|=\sqrt{a^2+b^2}$. Apply the Pythagorean theorem to find the length of the vector.
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What is the unit vector in the direction of a nonzero vector $\vec{v}$?
What is the unit vector in the direction of a nonzero vector $\vec{v}$?
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$\hat{v}=\frac{\vec{v}}{|\vec{v}|}$. Divide the vector by its magnitude to get a vector of length 1.
$\hat{v}=\frac{\vec{v}}{|\vec{v}|}$. Divide the vector by its magnitude to get a vector of length 1.
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State the formula for vector addition in components: $\langle a,b\rangle+\langle c,d\rangle$.
State the formula for vector addition in components: $\langle a,b\rangle+\langle c,d\rangle$.
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$\langle a+c,,b+d\rangle$. Add corresponding components to get the resultant vector.
$\langle a+c,,b+d\rangle$. Add corresponding components to get the resultant vector.
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State the formula for scalar multiplication: $k\langle a,b\rangle$.
State the formula for scalar multiplication: $k\langle a,b\rangle$.
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$\langle ka,,kb\rangle$. Multiply each component by the scalar $k$.
$\langle ka,,kb\rangle$. Multiply each component by the scalar $k$.
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What is the dot product formula for $\vec{u}=\langle a,b\rangle$ and $\vec{v}=\langle c,d\rangle$?
What is the dot product formula for $\vec{u}=\langle a,b\rangle$ and $\vec{v}=\langle c,d\rangle$?
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$\vec{u}\cdot\vec{v}=ac+bd$. Multiply corresponding components and add the products.
$\vec{u}\cdot\vec{v}=ac+bd$. Multiply corresponding components and add the products.
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What is the angle relation for dot product using magnitudes and angle $\theta$?
What is the angle relation for dot product using magnitudes and angle $\theta$?
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$\vec{u}\cdot\vec{v}=|\vec{u}|,|\vec{v}|\cos\theta$. Relates dot product to magnitudes and the angle between vectors.
$\vec{u}\cdot\vec{v}=|\vec{u}|,|\vec{v}|\cos\theta$. Relates dot product to magnitudes and the angle between vectors.
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What condition on $\vec{u}\cdot\vec{v}$ shows two nonzero vectors are perpendicular?
What condition on $\vec{u}\cdot\vec{v}$ shows two nonzero vectors are perpendicular?
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$\vec{u}\cdot\vec{v}=0$. Perpendicular vectors have a dot product of zero.
$\vec{u}\cdot\vec{v}=0$. Perpendicular vectors have a dot product of zero.
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What is the formula for the projection of $\vec{v}$ onto $\vec{u}$ (vector projection)?
What is the formula for the projection of $\vec{v}$ onto $\vec{u}$ (vector projection)?
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$\text{proj}_{\vec{u}}\vec{v}=\frac{\vec{v}\cdot\vec{u}}{|\vec{u}|^2},\vec{u}$. Projects $\vec{v}$ onto $\vec{u}$ using dot product and scaling.
$\text{proj}_{\vec{u}}\vec{v}=\frac{\vec{v}\cdot\vec{u}}{|\vec{u}|^2},\vec{u}$. Projects $\vec{v}$ onto $\vec{u}$ using dot product and scaling.
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What is the formula for the scalar component of $\vec{v}$ in the direction of $\vec{u}$?
What is the formula for the scalar component of $\vec{v}$ in the direction of $\vec{u}$?
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$\text{comp}_{\vec{u}}\vec{v}=\frac{\vec{v}\cdot\vec{u}}{|\vec{u}|}$. Gives the signed length of the projection of $\vec{v}$ onto $\vec{u}$.
$\text{comp}_{\vec{u}}\vec{v}=\frac{\vec{v}\cdot\vec{u}}{|\vec{u}|}$. Gives the signed length of the projection of $\vec{v}$ onto $\vec{u}$.
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What is the displacement vector after time $t$ with constant velocity $\vec{v}$?
What is the displacement vector after time $t$ with constant velocity $\vec{v}$?
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$\Delta\vec{r}=t\vec{v}$. Displacement equals velocity times time for constant motion.
$\Delta\vec{r}=t\vec{v}$. Displacement equals velocity times time for constant motion.
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What is the position vector formula for constant velocity: initial $\vec{r}_0$ and velocity $\vec{v}$?
What is the position vector formula for constant velocity: initial $\vec{r}_0$ and velocity $\vec{v}$?
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$\vec{r}(t)=\vec{r}_0+t\vec{v}$. Position equals initial position plus displacement over time.
$\vec{r}(t)=\vec{r}_0+t\vec{v}$. Position equals initial position plus displacement over time.
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Identify the speed if velocity is $\vec{v}=\langle v_x,v_y\rangle$.
Identify the speed if velocity is $\vec{v}=\langle v_x,v_y\rangle$.
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$\text{speed}=|\vec{v}|=\sqrt{v_x^2+v_y^2}$. Speed is the magnitude of the velocity vector.
$\text{speed}=|\vec{v}|=\sqrt{v_x^2+v_y^2}$. Speed is the magnitude of the velocity vector.
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Find the resultant velocity: $\langle 3, -2\rangle+\langle -5, 7\rangle$.
Find the resultant velocity: $\langle 3, -2\rangle+\langle -5, 7\rangle$.
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$\langle -2,,5\rangle$. Add components: $(3-5, -2+7) = \langle -2, 5\rangle$.
$\langle -2,,5\rangle$. Add components: $(3-5, -2+7) = \langle -2, 5\rangle$.
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Find the speed of $\vec{v}=\langle 6,8\rangle$.
Find the speed of $\vec{v}=\langle 6,8\rangle$.
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$10$. $|\vec{v}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.
$10$. $|\vec{v}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.
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Find a unit vector in the direction of $\vec{v}=\langle 3,4\rangle$.
Find a unit vector in the direction of $\vec{v}=\langle 3,4\rangle$.
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$\left\langle \frac{3}{5},,\frac{4}{5}\right\rangle$. $|\vec{v}| = 5$, so $\hat{v} = \frac{1}{5}\langle 3,4\rangle$.
$\left\langle \frac{3}{5},,\frac{4}{5}\right\rangle$. $|\vec{v}| = 5$, so $\hat{v} = \frac{1}{5}\langle 3,4\rangle$.
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Compute the dot product $\langle 2,-1\rangle\cdot\langle 5,4\rangle$.
Compute the dot product $\langle 2,-1\rangle\cdot\langle 5,4\rangle$.
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$6$. $(2)(5) + (-1)(4) = 10 - 4 = 6$.
$6$. $(2)(5) + (-1)(4) = 10 - 4 = 6$.
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Find $\text{proj}_{\vec{u}}\vec{v}$ for $\vec{u}=\langle 1,0\rangle$ and $\vec{v}=\langle 3,4\rangle$.
Find $\text{proj}_{\vec{u}}\vec{v}$ for $\vec{u}=\langle 1,0\rangle$ and $\vec{v}=\langle 3,4\rangle$.
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$\langle 3,,0\rangle$. $\frac{\vec{v}\cdot\vec{u}}{|\vec{u}|^2}\vec{u} = \frac{3}{1}\langle 1,0\rangle = \langle 3,0\rangle$.
$\langle 3,,0\rangle$. $\frac{\vec{v}\cdot\vec{u}}{|\vec{u}|^2}\vec{u} = \frac{3}{1}\langle 1,0\rangle = \langle 3,0\rangle$.
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Find position at $t=3$ if $\vec{r}_0=\langle 2,-1\rangle$ and $\vec{v}=\langle -4,5\rangle$.
Find position at $t=3$ if $\vec{r}_0=\langle 2,-1\rangle$ and $\vec{v}=\langle -4,5\rangle$.
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$\langle -10,,14\rangle$. $\vec{r}(3) = \langle 2,-1\rangle + 3\langle -4,5\rangle = \langle 2-12, -1+15\rangle$.
$\langle -10,,14\rangle$. $\vec{r}(3) = \langle 2,-1\rangle + 3\langle -4,5\rangle = \langle 2-12, -1+15\rangle$.
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What is the distance traveled in time $t$ at constant velocity vector $\vec{v}$ (assume $t\ge 0$)?
What is the distance traveled in time $t$ at constant velocity vector $\vec{v}$ (assume $t\ge 0$)?
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$\text{distance}=|\vec{v}|t$. Speed times time equals distance traveled.
$\text{distance}=|\vec{v}|t$. Speed times time equals distance traveled.
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Find the magnitude of $\vec{v}=\langle 3,4\rangle$.
Find the magnitude of $\vec{v}=\langle 3,4\rangle$.
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$5$. $\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$.
$5$. $\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$.
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Compute the dot product $\langle 2,-1\rangle\cdot\langle 3,4\rangle$.
Compute the dot product $\langle 2,-1\rangle\cdot\langle 3,4\rangle$.
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$2$. $(2)(3)+(-1)(4)=6-4=2$.
$2$. $(2)(3)+(-1)(4)=6-4=2$.
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Find $\operatorname{proj}_{\langle 1,0\rangle}\langle 3,4\rangle$.
Find $\operatorname{proj}_{\langle 1,0\rangle}\langle 3,4\rangle$.
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$\langle 3,0\rangle$. Projects onto x-axis: keeps x-component, zeros y.
$\langle 3,0\rangle$. Projects onto x-axis: keeps x-component, zeros y.
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Find the resultant of perpendicular velocities $\langle 5,0\rangle$ and $\langle 0,12\rangle$.
Find the resultant of perpendicular velocities $\langle 5,0\rangle$ and $\langle 0,12\rangle$.
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$\langle 5,12\rangle$. Add components: $\langle 5+0,0+12\rangle=\langle 5,12\rangle$.
$\langle 5,12\rangle$. Add components: $\langle 5+0,0+12\rangle=\langle 5,12\rangle$.
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Find the speed (magnitude) of the velocity vector $\langle 5,12\rangle$.
Find the speed (magnitude) of the velocity vector $\langle 5,12\rangle$.
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$13$. $\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$.
$13$. $\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$.
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With $\vec{r}_0=\langle 1,2\rangle$ and $\vec{v}=\langle 3,-1\rangle$, what is $\vec{r}(4)$?
With $\vec{r}_0=\langle 1,2\rangle$ and $\vec{v}=\langle 3,-1\rangle$, what is $\vec{r}(4)$?
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$\langle 13,-2\rangle$. $\langle 1,2\rangle+4\langle 3,-1\rangle=\langle 1+12,2-4\rangle$.
$\langle 13,-2\rangle$. $\langle 1,2\rangle+4\langle 3,-1\rangle=\langle 1+12,2-4\rangle$.
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What is the component form of the displacement vector from $(x_1,y_1)$ to $(x_2,y_2)$?
What is the component form of the displacement vector from $(x_1,y_1)$ to $(x_2,y_2)$?
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$\langle x_2-x_1,\ y_2-y_1\rangle$. Subtract initial from terminal coordinates to get displacement.
$\langle x_2-x_1,\ y_2-y_1\rangle$. Subtract initial from terminal coordinates to get displacement.
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A boat has velocity $\langle 4,0\rangle$ in still water; current is $\langle 0,3\rangle$. Find ground velocity.
A boat has velocity $\langle 4,0\rangle$ in still water; current is $\langle 0,3\rangle$. Find ground velocity.
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$\langle 4,3\rangle$. Add boat and current vectors component-wise.
$\langle 4,3\rangle$. Add boat and current vectors component-wise.
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State the formula for the magnitude of a $2$D vector $\vec{v}=\langle a,b\rangle$.
State the formula for the magnitude of a $2$D vector $\vec{v}=\langle a,b\rangle$.
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$|\vec{v}|=\sqrt{a^2+b^2}$. Apply the Pythagorean theorem to vector components.
$|\vec{v}|=\sqrt{a^2+b^2}$. Apply the Pythagorean theorem to vector components.
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