Question

find each quantity if $v=3i-5j$and $w=-2i+3j$

$\left|v-w\right|$

Short answer

The value of $\left|\left|v-w\right|\right|$when $v=3i-5j$and $w=-2i+3j$is$\sqrt{89}$

Step-by-step solution

Step 1. Given data

$v=3i-5j$and$w=-2i+3j$

Step 2. Find the exact valueSubstitute the values of v and w in the equation src="data:image/svg+xml;base64,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" role="math" localid="1646674606882" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/c4ca47a8-48d3-418c-ab14-2c93585cf0a7.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220309%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220309T055122Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=6876e6afc8f8bc5066875d5678767614b46281bc1370184ba5fac6483b45d850" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/c4ca47a8-48d3-418c-ab14-2c93585cf0a7.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220308%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220308T070015Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=e76271610ac9c23cdf01afb053d572194191c8dd8d8dc9719fffabcdcc3aae3f" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/c4ca47a8-48d3-418c-ab14-2c93585cf0a7.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220307%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220307T174644Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=c0a3484104f05f35b49b15ba4f0e465edffd5dc3aa683ff2cf5a8ff5edee54e1" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/c4ca47a8-48d3-418c-ab14-2c93585cf0a7.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220307%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220307T174240Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=6c7e631c65814f4c350d116216379ac7969d588950f97bbfbe8ac6ee140f1a60" v-w

$v-w=3i-5j-(-2i+3j)$

$v-w=3i-5j+2i-3j$

$v-w=5i-8j$

Step 3. Find the magnitude

$\left|v-w\right|=\sqrt{5^2+8^2}$

$\left|v-w\right|=\sqrt{25+64}$

$\left|v-w\right|=\sqrt{89}$