Name: Resolution of Vector Into Components
Uploaded: 2019-07-19
Duration: 8 min 23 s
Description: The resolution of a vector means splitting of a vector into two or more directions. Let us learn how to resolve a vector through this video.

Question

Let

\vec{a}, \vec{b}, \vec{c}

be unit vectors such that

\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0

and the angle between

\vec{b}

and

\vec{c}

is

\frac{π}{3}

. Prove that

\vec{a} = \pm \frac{2}{\sqrt{3}} (\vec{b} \times \vec{c})

.

Accepted Answer

Given that, $\vec{a}, \vec{b}, \vec{c}$ are unit vectors.

$∴ | \vec{a} | = 1, | \vec{b} | = 1, | \vec{c} | = 1$

Also $\vec{a} \cdot \vec{b} = 0, \vec{a} \cdot \vec{c} = 0$

Angle between $\vec{b}$ and $\vec{c} = \frac{π}{3}$

$\vec{a} \cdot \vec{b} = 0 \Rightarrow \begin{array}{l} \vec{a} & ⊥^{r} & \vec{b} \end{array}$

$\vec{a} \cdot \vec{c} = 0 \begin{array}{l} \Rightarrow & \vec{a} & ⊥^{r} & \vec{c} \end{array}$

$∴ \vec{a}$ is perpendicular to both $\vec{b}$ and $\vec{c}$ .

$\vec{b} \times \vec{c} = | \vec{b} | | \vec{c} | \sin \frac{π}{3} \hat{n}$

Where $\hat{n}$ is a unit vector perpendicular to both $\vec{b}$ and $\vec{c}$ which is $\vec{a}$ .

$\vec{b} \times \vec{c} = 1 \times 1 \times \frac{\sqrt{3}}{2} \times \hat{n}$

$\pm \frac{2}{\sqrt{3}} (\vec{b} \times \vec{c}) = \pm \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{2} \times \hat{n}$

$\pm \frac{2}{\sqrt{3}} (\vec{b} \times \vec{c}) = \pm \hat{n} \dots \dots (1)$

$\pm \hat{n}$ is a unit vector perpendicular to both $\vec{b}$ and $\vec{c}$ which is $\vec{a}$

$(1) \Rightarrow \pm \frac{2}{\sqrt{3}} (\vec{b} \times \vec{c}) = \vec{a}$

$\vec{a} = \pm \frac{2}{\sqrt{3}} (\vec{b} \times \vec{c})$ .

Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{π}{3}$ . Prove that $\vec{a} = \pm \frac{2}{\sqrt{3}} (\vec{b} \times \vec{c})$ .

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

Let a→, b→, c→ be unit vectors such that a→·b→=a→·c→=0 and the angle between b→ and c→ is π3. Prove that a→=±23(b→×c→).

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{π}{3}$ . Prove that $\vec{a} = \pm \frac{2}{\sqrt{3}} (\vec{b} \times \vec{c})$ .