Find the values of x and y so that vectors a = 2i + 3j, b = xi + yj are orthogonal.

Tunapewa vekta:

$$\vec{a} = 2i + 3j$$ $$\vec{b} = xi + yj$$

Vekta $\vec{a}$ na $\vec{b}$ ni orthogonal ikiwa dot product yao ni sifuri:

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

Dot product inakokotolewa kwa kuzidisha vijenzi vinavyolingana na kisha kujumlisha matokeo:

$$\vec{a} \cdot \vec{b} = (2)(x) + (3)(y)$$ $$\vec{a} \cdot \vec{b} = 2x + 3y$$

Kwa kuwa vekta ni orthogonal, tunapata mlinganyo:

$$2x + 3y = 0$$

Kutoka kwenye mlinganyo $2x + 3y = 0$, tunaweza kueleza $x$ kwa kutumia $y$ au $y$ kwa kutumia $x$.

$$2x = -3y$$ $$x = -\frac{3}{2}y$$

Au

$$3y = -2x$$ $$y = -\frac{2}{3}x$$

Kuna jozi nyingi zisizo na kikomo za $(x, y)$ zinazokidhi masharti haya. Kwa mfano, ikiwa $y=2$, basi $x = -\frac{3}{2}(2) = -3$. Hivyo, $\vec{b} = -3i + 2j$ ingekuwa orthogonal kwa $\vec{a}$.

Thamani za $x$ na $y$ lazima zikidhi uhusiano: