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Find the value t such that vectors a = 2i − 5j + k and b = -3i + tj - 7k are perpendicular (a.b = 0)

June 7, 2026Mathematics

ScanSolve AI Answer

Answer

-\frac{13}{5}

Step-by-step solution

3 steps

To find the value of $t$ such that vectors $\vec{a}$ and $\vec{b}$ are perpendicular, we use the condition that their dot product must be zero ( $\vec{a} \cdot \vec{b} = 0$ ).

Write down the given vectors.
$\vec{a} = 2\vec{i} - 5\vec{j} + \vec{k}$ $\vec{b} = -3\vec{i} + t\vec{j} - 7\vec{k}$
Calculate the dot product $\vec{a} \cdot \vec{b}$ .
The dot product is found by multiplying the corresponding components and summing them.
$\vec{a} \cdot \vec{b} = (2)(-3) + (-5)(t) + (1)(-7)$
Set the dot product equal to zero and solve for $t$ .
$-6 - 5t - 7 = 0$ $-13 - 5t = 0$ $-5t = 13$ $t = -\frac{13}{5}$
The value of $t$ for which the vectors are perpendicular is $\boxed{-\frac{13}{5}}$ .