Geometry

Cross Product of Vectors

Cross Product of Vectors: Level 3 Challenges

         

Consider the force vector F=⟨1,1,12⟩\textbf{F} = \langle1, 1, \frac{1}{2}\rangle . If the magnitude of the torque τ=OP‾×F\tau = \overline{OP}\times\textbf{F} is equal to the area of the equilateral triangle formed by the origin, PP, and (1,−1,1)(1, -1, 1), then determine the acute angle formed by OP‾\overline{OP} and F\textbf{F}. Give your answer in degrees.

Andrew Ellinor by Andrew Ellinor

Let a⃗,b⃗\vec{a},\vec{b} and c⃗\vec{c} be three non-zero vectors such that no two of them are collinear and (a⃗×b⃗)×c⃗=13∣b⃗∣∣c⃗∣a⃗(\vec{a} \times \vec{b}) \times \vec{c}=\frac{1}{3} |\vec{b}||\vec{c}| \vec{a}. If θ\theta is the angle between vectors b⃗\vec{b} and c⃗\vec{c}, then the value of sin⁡θ\sin\theta is :

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Sandeep Bhardwaj by Sandeep Bhardwaj

Let u^\hat{u} and v^\hat{v} be unit vectors and w⃗\vec{w} be a vector such that w⃗+(w⃗ ×u^)\vec{w}+(\vec{w}\ \times \hat{u}) == v^\hat{v}.

The angle in degrees between u^\hat{u} and v^\hat{v} such that ∣(u^×v^)⋅w⃗∣|(\hat{u} \times \hat{v}) \cdot \vec{w}| is maximized is θ\theta and the maximum value of ∣(u^×v^)⋅w⃗∣|(\hat{u} \times \hat{v}) \cdot \vec{w}| is MM. Find the value of θ+M\theta + M.

ImageImage Credit:Credit : WikipediaWikipedia

Led Tasso by Led Tasso

Suppose p⃗,q⃗, and r⃗\vec{p} , \vec{q}, \text{ and } \vec{r} are three mutually perpendicular unit vectors.

Vector u⃗\vec{u} satisfies the equation p⃗×((u⃗−q⃗)×p⃗)+q⃗×((u⃗−r⃗)×q⃗)+r⃗×((u⃗−p⃗)×r⃗)=0\vec{p}\times((\vec{u} - \vec{q})\times\vec{p}) \hspace{.15cm} + \hspace{.15cm} \vec{q}\times((\vec{u} - \vec{r})\times\vec{q}) \hspace{.15cm} + \hspace{.15cm}\vec{r}\times((\vec{u} - \vec{p})\times\vec{r}) = 0

What is u⃗\vec{u} in terms of p⃗,q⃗, and r⃗\vec{p} , \vec{q}, \text{ and } \vec{r}?

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Harshit Damani by Harshit Damani

Consider the regular octagon centered at the origin as shown at right. Eight unit vectors are drawn from the center of the octagon to each of its vertices and labeled in the figure. For each pair of distinct unit vectors u⃗i,u⃗j\vec{u}_i, \vec{u}_j with i<ji < j, their cross product is computed.

What is the sum of all of these cross products?

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Andrew Ellinor by Andrew Ellinor
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